🧠:<think>Okay, so I have this problem where I need to help a homemaker plan a week's worth of meals for a household of four, each with different dietary needs. The goal is to minimize the total cost of the meal plan while meeting certain nutritional requirements. There are two sub-problems here, and I need to tackle them one by one.Starting with Sub-problem 1: Formulate an optimization problem to minimize the total cost. The constraints are that each member needs 50g of protein, 200g of carbs, and 70g of fat daily. The costs are 0.03 per gram for protein, 0.02 for carbs, and 0.05 for fat. Also, the total weekly intake for each nutrient should be within 5% of the recommended daily intake multiplied by 7.First, let me understand the recommended weekly intake. For each nutrient, it's 7 times the daily recommendation. So, protein would be 50*7=350g, carbs 200*7=1400g, and fat 70*7=490g. But the total weekly intake should be within 5% of these amounts. That means the total protein can be between 350*(1-0.05)=332.5g and 350*(1+0.05)=367.5g. Similarly for carbs and fat.Now, the matrix N is given as a 7x3 matrix where each row represents a day, and the columns are protein, carbs, fat. So, each day has P_i, C_i, F_i grams of each nutrient.The goal is to choose how much of each nutrient to consume each day such that the total cost is minimized, while meeting the weekly intake constraints.Wait, but the matrix N is given. Does that mean that the nutrients per day are fixed? Or is N a variable that we can adjust? Hmm, the problem says "given the matrix N", so I think N is fixed. So, we can't change the amount of each nutrient per day. Then, how do we adjust the intake? Maybe the homemaker can choose how many servings or portions to prepare each day, scaling up or down the nutrients accordingly.But the problem doesn't specify that. It just says "given the matrix N". Hmm, maybe I need to think differently. Perhaps the matrix N is the amount of each nutrient provided by the meal plan each day, and we need to adjust the quantities to meet the weekly requirements.Wait, maybe the homemaker can adjust the portions each day, so that the total weekly intake is within the 5% range. So, for each day, we can scale the nutrients up or down by a certain factor, but keeping the ratio of protein, carbs, and fat the same each day.Wait, that sounds more like Sub-problem 2. Let me check.Sub-problem 2 says: Given the matrix N, develop an algorithm to balance the nutritional intake such that each day's nutrient distribution remains the same, but the total weekly intake is adjusted to meet the household’s dietary needs while minimizing the difference in daily intake from the original plan. Apply linear programming techniques.Okay, so Sub-problem 2 is about scaling each day's nutrients by a factor so that the total weekly intake meets the household's needs, while keeping the daily ratios the same. And it wants to minimize the difference from the original plan.But Sub-problem 1 is about formulating an optimization problem to minimize total cost, given the constraints on weekly intake. So, perhaps in Sub-problem 1, the nutrients per day are variables, and we need to choose them such that the total cost is minimized, while meeting the weekly intake constraints.Wait, but the matrix N is given. Maybe N is the amount of each nutrient provided by the meal plan each day, and we need to decide how much of each meal to prepare each day, scaling the nutrients accordingly, while keeping the ratios the same each day.Wait, I'm getting confused. Let me read the problem again.The matrix N represents the nutritional content (in grams) of various essential nutrients (Protein, Carbohydrates, and Fat) for each of the 7 days of the week. So, each day has P_i, C_i, F_i grams.Sub-problem 1: Formulate an optimization problem to minimize the total cost, given constraints on daily intake, cost per gram, and total weekly intake within 5% of recommended.So, perhaps the homemaker can choose how much of each day's meal to prepare, scaling the nutrients accordingly. So, for each day, we can have a variable x_i representing the scaling factor for day i. Then, the total protein would be sum over i of x_i * P_i, similarly for carbs and fat.The cost would be sum over i of (0.03*P_i + 0.02*C_i + 0.05*F_i) * x_i.We need to minimize this cost, subject to:sum(x_i * P_i) >= 332.5 and <= 367.5sum(x_i * C_i) >= 1330 (1400*0.95) and <= 1470 (1400*1.05)sum(x_i * F_i) >= 465.5 (490*0.95) and <= 514.5 (490*1.05)Also, x_i >= 0 for all i.Additionally, since each day's meal should be prepared in a way that the ratios of nutrients are maintained, but actually, in Sub-problem 1, it's just about the total weekly intake, so maybe the ratios don't matter, only the totals.Wait, but the problem says "the total intake for each nutrient over the week should be within 5% of the recommended daily intake multiplied by 7." So, that's just the totals, not the daily intake.But each member has different dietary needs, but the problem says the recommended daily intake is 50g protein, 200g carbs, 70g fat. So, for four members, the total daily intake would be 4*50=200g protein, 4*200=800g carbs, 4*70=280g fat. Therefore, weekly intake would be 1400g protein, 5600g carbs, 1960g fat.Wait, hold on. That might be a misinterpretation. The problem says "the recommended daily intake for each member is 50g protein, 200g carbs, 70g fat." So, for four members, the total daily intake would be 4*50=200g protein, 4*200=800g carbs, 4*70=280g fat. Therefore, weekly intake would be 200*7=1400g protein, 800*7=5600g carbs, 280*7=1960g fat.But the problem says the total weekly intake should be within 5% of the recommended daily intake multiplied by 7. So, the recommended daily intake per member is 50,200,70. For four members, the daily intake is 200,800,280. Weekly is 1400,5600,1960. So, the total weekly intake should be within 5% of these.So, the total protein should be between 1400*0.95=1330 and 1400*1.05=1470.Similarly, carbs: 5600*0.95=5320 to 5600*1.05=5880.Fat: 1960*0.95=1862 to 1960*1.05=2058.So, the constraints are:sum(x_i * P_i) ∈ [1330, 1470]sum(x_i * C_i) ∈ [5320, 5880]sum(x_i * F_i) ∈ [1862, 2058]And x_i >= 0 for all i.The objective is to minimize the total cost, which is sum(x_i * (0.03*P_i + 0.02*C_i + 0.05*F_i)).So, that's the optimization problem.But wait, the problem says "the total intake for each nutrient over the week should be within 5% of the recommended daily intake multiplied by 7." So, it's 5% of the weekly recommended intake, not 5% of the daily multiplied by 7. Wait, no, the wording is "within 5% of the recommended daily intake multiplied by 7." So, that would be 5% of (50*7)=350, so 350±17.5, which is 332.5 to 367.5 for protein. Similarly, carbs: 200*7=1400, so 1400±70=1330 to 1470. Fat: 70*7=490, so 490±24.5=465.5 to 514.5.Wait, but earlier I thought it's per member, so for four members, it's 4 times that. So, I'm confused now.Wait, the problem says "the recommended daily intake for each member is 50g protein, 200g carbs, 70g fat." So, per member, daily. So, for four members, the total daily intake is 4*50=200g protein, 4*200=800g carbs, 4*70=280g fat. Therefore, weekly intake is 200*7=1400g protein, 800*7=5600g carbs, 280*7=1960g fat.But the problem says the total weekly intake should be within 5% of the recommended daily intake multiplied by 7. So, for protein, it's 50*7=350g, so 350±17.5=332.5 to 367.5. But that's per member? Or total?Wait, no. The wording is ambiguous. It says "the total intake for each nutrient over the week should be within 5% of the recommended daily intake multiplied by 7."So, "recommended daily intake" is per member, so 50g protein per day per member. So, for four members, the total daily intake is 200g protein. Therefore, the weekly intake is 1400g. But the problem says the total weekly intake should be within 5% of (recommended daily intake *7). So, 50*7=350g, so 350±17.5=332.5 to 367.5. But that's per member? Or total?Wait, no, the total intake for the household is 4*50=200g protein per day, so 1400g per week. The problem says the total weekly intake should be within 5% of (recommended daily intake *7). So, 50*7=350g, so 350±17.5=332.5 to 367.5. But that's per member? Or total?Wait, I think it's per member. Because the recommended daily intake is per member. So, the total weekly intake per member should be within 5% of 50*7=350g. So, for each member, their weekly protein intake should be between 332.5 and 367.5g. Therefore, for four members, the total weekly protein intake should be between 4*332.5=1330g and 4*367.5=1470g.Similarly, carbs: 200*7=1400g per member weekly, so total 4*1400=5600g. Within 5%: 5600±280=5320 to 5880.Fat: 70*7=490g per member weekly, total 4*490=1960g. Within 5%: 1960±98=1862 to 2058.So, that aligns with my initial thought. So, the total weekly intake for the household should be within 5% of 4*(recommended daily intake)*7.Therefore, the constraints are:sum(x_i * P_i) ∈ [1330, 1470]sum(x_i * C_i) ∈ [5320, 5880]sum(x_i * F_i) ∈ [1862, 2058]And x_i >= 0 for all i.The objective is to minimize the total cost, which is sum(x_i * (0.03*P_i + 0.02*C_i + 0.05*F_i)).So, that's the optimization problem.Now, for Sub-problem 2, it's about adjusting the weekly intake to meet the household's dietary needs while minimizing the difference from the original plan. It mentions that each day's nutrient distribution remains the same, so the ratios of P:C:F are kept per day, but the total weekly intake is adjusted.So, perhaps we can model this as scaling each day's meal by a factor, say s_i, such that the total weekly intake meets the household's needs, while minimizing the sum of squared differences between the scaled and original meals.But the problem says to apply linear programming techniques, so maybe it's a linear approximation.Alternatively, since the ratios must remain the same, we can scale each day's meal by a factor, but the scaling factor can vary per day, as long as the ratios are maintained.Wait, but if the ratios are maintained, then the scaling factor for each day would be the same across all nutrients. So, for each day i, we have s_i such that the scaled nutrients are s_i*P_i, s_i*C_i, s_i*F_i.Then, the total weekly intake would be sum(s_i*P_i), sum(s_i*C_i), sum(s_i*F_i).We need these totals to meet the household's needs, which are 1400g protein, 5600g carbs, 1960g fat, but perhaps adjusted to be within the 5% range as in Sub-problem 1.But the problem says "to meet the household’s dietary needs while minimizing the difference in daily intake from the original plan." So, perhaps the household's needs are exactly 1400,5600,1960, and we need to scale the meals so that the totals are exactly these amounts, while minimizing the difference from the original plan.But the problem says "minimizing the difference in daily intake from the original plan." So, maybe we need to minimize the sum of |s_i*P_i - P_i| + |s_i*C_i - C_i| + |s_i*F_i - F_i| over all days, subject to sum(s_i*P_i)=1400, sum(s_i*C_i)=5600, sum(s_i*F_i)=1960.But that's a linear programming problem if we use linear approximations for the absolute differences.Alternatively, we can use a quadratic objective function, but since the problem specifies linear programming, we need to linearize the absolute values.Alternatively, perhaps we can minimize the sum of squared differences, but that would be quadratic, not linear.Wait, but the problem says "minimizing the difference in daily intake from the original plan." So, perhaps it's the sum of the absolute differences between the scaled and original nutrients per day.But since linear programming can't handle absolute values directly, we can introduce auxiliary variables and constraints to linearize it.Alternatively, maybe the problem expects us to scale each day's meal by the same factor, so that the ratios are maintained across all days, but that might not be necessary.Wait, the problem says "each day's nutrient distribution remains the same," which means that for each day, the ratio of P:C:F is maintained. So, for each day i, the scaling factor s_i is the same for P_i, C_i, F_i. So, s_i is a scalar multiplier for each day.Therefore, the total weekly intake would be sum(s_i*P_i), sum(s_i*C_i), sum(s_i*F_i).We need these totals to meet the household's needs, which are 1400,5600,1960.But the problem says "to meet the household’s dietary needs while minimizing the difference in daily intake from the original plan."So, perhaps the household's needs are exactly 1400,5600,1960, and we need to find s_i such that sum(s_i*P_i)=1400, sum(s_i*C_i)=5600, sum(s_i*F_i)=1960, and minimize the sum over days of (s_i*P_i - P_i)^2 + (s_i*C_i - C_i)^2 + (s_i*F_i - F_i)^2.But that's a quadratic optimization problem. However, the problem specifies to apply linear programming techniques, so maybe we need to linearize this.Alternatively, perhaps we can use a linear approximation of the difference, such as the sum of absolute differences, but that would require introducing auxiliary variables and constraints.Alternatively, maybe the problem expects us to scale each day's meal by the same factor, so that s_i = s for all i, which would make the problem linear. But that might not be necessary, as the problem only requires that each day's nutrient distribution remains the same, not that the scaling factor is the same across days.Wait, but if each day's nutrient distribution remains the same, then the ratios P_i:C_i:F_i are the same for each day, but the scaling factors s_i can vary per day. So, it's possible to have different s_i for each day, as long as the ratios are maintained.But then, the problem becomes a linear system with variables s_i, and constraints sum(s_i*P_i)=1400, sum(s_i*C_i)=5600, sum(s_i*F_i)=1960, and s_i >=0.But that's a system of three equations with seven variables, so it's underdetermined. Therefore, we need to add an objective function to minimize the difference from the original plan.So, the objective is to minimize the sum over days of |s_i*P_i - P_i| + |s_i*C_i - C_i| + |s_i*F_i - F_i|.But since this is linear programming, we can't have absolute values. So, we can introduce variables d_i for each nutrient and day, such that d_i = s_i*P_i - P_i, and similarly for C and F, and then minimize the sum of d_i's absolute values.But in linear programming, we can't have absolute values, so we can split each d_i into positive and negative parts, d_i^+ and d_i^-, such that d_i = d_i^+ - d_i^-, and |d_i| = d_i^+ + d_i^-.Then, the objective becomes minimizing the sum over all nutrients and days of (d_i^+ + d_i^-).Subject to:sum(s_i*P_i) = 1400sum(s_i*C_i) = 5600sum(s_i*F_i) = 1960s_i*P_i - P_i = d_i^+ - d_i^-s_i*C_i - C_i = e_i^+ - e_i^-s_i*F_i - F_i = f_i^+ - f_i^-d_i^+, d_i^-, e_i^+, e_i^-, f_i^+, f_i^- >=0But this is getting complicated, but it's a standard way to linearize absolute values.Alternatively, since the problem mentions "minimizing the difference in daily intake from the original plan," perhaps it's acceptable to minimize the sum of squared differences, but that would be quadratic, not linear.Alternatively, maybe the problem expects a simpler approach, such as scaling each day's meal by the same factor s, so that s*sum(P_i)=1400, s*sum(C_i)=5600, s*sum(F_i)=1960.But that would require that sum(P_i)=1400/s, sum(C_i)=5600/s, sum(F_i)=1960/s, which might not be possible unless the ratios of sum(P_i):sum(C_i):sum(F_i) are the same as 1400:5600:1960, which simplifies to 1:4:14.But unless the original matrix N has sum(P_i):sum(C_i):sum(F_i) =1:4:14, this scaling factor s would not work. So, perhaps the problem expects us to scale each day's meal by a different factor s_i, such that the total weekly intake meets the requirements, while minimizing the sum of the absolute differences.But given the complexity, perhaps the problem expects us to scale each day's meal by the same factor s, so that s*sum(P_i)=1400, s*sum(C_i)=5600, s*sum(F_i)=1960.But let's check if that's possible.Let me denote S_P = sum(P_i), S_C = sum(C_i), S_F = sum(F_i).Then, s = 1400/S_P, and also s = 5600/S_C, and s = 1960/S_F.Therefore, for this to be possible, 1400/S_P = 5600/S_C = 1960/S_F.Which implies that S_P/S_C = 1400/5600 = 1/4, and S_P/S_F = 1400/1960 = 5/7.So, unless the original matrix N has these ratios, this scaling won't work. Therefore, the problem likely expects us to scale each day's meal by a different factor s_i, so that the total weekly intake meets the requirements, while minimizing the sum of the absolute differences.Therefore, the linear programming formulation would involve variables s_i for each day, and auxiliary variables to represent the absolute differences, as I outlined earlier.But perhaps a simpler approach is to minimize the sum of (s_i -1)^2, but that's quadratic. Alternatively, minimize the sum of |s_i -1|, which is linear if we use auxiliary variables.So, to summarize, for Sub-problem 2, the algorithm would involve:1. Introduce scaling factors s_i for each day i.2. Ensure that sum(s_i*P_i)=1400, sum(s_i*C_i)=5600, sum(s_i*F_i)=1960.3. Minimize the sum over days of |s_i*P_i - P_i| + |s_i*C_i - C_i| + |s_i*F_i - F_i|.But to linearize the absolute values, we can introduce variables for the positive and negative deviations and add them to the objective.Alternatively, since the problem mentions "minimizing the difference in daily intake from the original plan," perhaps it's acceptable to minimize the sum of the absolute differences between the scaled and original nutrients per day.So, the linear programming formulation would be:Minimize sum_{i=1 to 7} [ (s_i*P_i - P_i)^+ + (s_i*P_i - P_i)^- + (s_i*C_i - C_i)^+ + (s_i*C_i - C_i)^- + (s_i*F_i - F_i)^+ + (s_i*F_i - F_i)^- ]Subject to:sum_{i=1 to 7} s_i*P_i = 1400sum_{i=1 to 7} s_i*C_i = 5600sum_{i=1 to 7} s_i*F_i = 1960s_i >=0 for all iAnd for each nutrient and day:s_i*P_i - P_i = (s_i*P_i - P_i)^+ - (s_i*P_i - P_i)^-Similarly for C and F.But this is a bit involved, but it's a standard way to handle absolute values in linear programming.Alternatively, since the problem mentions "minimizing the difference in daily intake from the original plan," perhaps it's acceptable to minimize the sum of the absolute differences between the scaled and original nutrients per day, which would be:Minimize sum_{i=1 to 7} [ |s_i*P_i - P_i| + |s_i*C_i - C_i| + |s_i*F_i - F_i| ]Subject to:sum_{i=1 to 7} s_i*P_i = 1400sum_{i=1 to 7} s_i*C_i = 5600sum_{i=1 to 7} s_i*F_i = 1960s_i >=0 for all iBut as mentioned, this requires linearizing the absolute values.Alternatively, perhaps the problem expects us to use a different approach, such as minimizing the maximum difference across days, but that would be a different objective.In any case, the key idea is to scale each day's meal by a factor s_i, maintaining the nutrient ratios per day, such that the total weekly intake meets the household's needs, while minimizing the deviation from the original plan.So, to formulate this as a linear program, we can proceed as follows:Let s_i be the scaling factor for day i.Define variables:For each day i and nutrient j (P, C, F), let d_{i,j} = s_i*N_{i,j} - N_{i,j}.We need to minimize sum_{i,j} |d_{i,j}|.Subject to:sum_{i} s_i*N_{i,P} = 1400sum_{i} s_i*N_{i,C} = 5600sum_{i} s_i*N_{i,F} = 1960s_i >=0 for all iBut since we can't have absolute values, we introduce variables d_{i,j}^+ and d_{i,j}^- such that:d_{i,j} = d_{i,j}^+ - d_{i,j}^-and |d_{i,j}| = d_{i,j}^+ + d_{i,j}^-So, the objective becomes:Minimize sum_{i,j} (d_{i,j}^+ + d_{i,j}^-)Subject to:sum_{i} s_i*N_{i,P} = 1400sum_{i} s_i*N_{i,C} = 5600sum_{i} s_i*N_{i,F} = 1960For each i,j:s_i*N_{i,j} - N_{i,j} = d_{i,j}^+ - d_{i,j}^-d_{i,j}^+, d_{i,j}^- >=0s_i >=0This is a linear program with variables s_i, d_{i,j}^+, d_{i,j}^-.So, that's the formulation for Sub-problem 2.In summary, for Sub-problem 1, the optimization problem is to choose scaling factors x_i for each day's meal to minimize the total cost, subject to the total weekly intake constraints. For Sub-problem 2, it's to scale each day's meal by a factor s_i to meet the exact weekly intake requirements while minimizing the deviation from the original plan, using linear programming with auxiliary variables for the absolute differences.</think>

🧠:<think>Alright, so I have these two probability questions to solve, both related to a colleague who's a dedicated gamer. Let me try to figure them out step by step.Starting with the first question: It says that the probability distribution of the gaming hours per day follows a normal distribution with a mean of 4 hours and a standard deviation of 1 hour. I need to calculate the probability that the gamer spends more than 5 hours gaming on a given day.Hmm, okay. So, normal distribution, mean (μ) is 4, standard deviation (σ) is 1. We need P(X > 5). Since it's a normal distribution, I can use the Z-score formula to standardize this value and then use the standard normal distribution table to find the probability.The Z-score formula is Z = (X - μ) / σ. Plugging in the numbers: Z = (5 - 4) / 1 = 1. So, the Z-score is 1. Now, I need to find the probability that Z is greater than 1. From the standard normal distribution table, the area to the left of Z=1 is approximately 0.8413. Therefore, the area to the right of Z=1, which is what we need, is 1 - 0.8413 = 0.1587. So, the probability is about 15.87%.Wait, let me double-check. If the mean is 4 and standard deviation is 1, then 5 is exactly one standard deviation above the mean. In a normal distribution, about 68% of the data lies within one standard deviation of the mean. So, 34% is between the mean and one standard deviation above, and 34% is between the mean and one standard deviation below. Therefore, the area above one standard deviation should be 16%, which aligns with the 0.1587 I calculated earlier. Okay, that seems consistent.Moving on to the second question: The gamer wants to rekindle their friend's interest by proposing a collaborative gaming session. The friend will only join if the total time spent gaming together in a week is within one standard deviation above or below the average weekly gaming hours of the dedicated gamer. I need to find the range of total gaming hours for which the friend agrees to join.First, let me parse this. The friend's availability follows a normal distribution similar to the gamer's daily gaming hours but scaled up to weekly hours. So, the daily distribution is normal with μ=4 and σ=1. If we scale this up to weekly hours, assuming the gamer plays every day, the weekly hours would be the sum of seven daily hours.Wait, is that correct? So, if each day is a normal variable with mean 4 and standard deviation 1, then the total weekly hours G would be the sum of seven such variables. The sum of independent normal variables is also normal, with mean equal to the sum of the means and variance equal to the sum of the variances.So, the weekly mean μ_weekly = 7 * 4 = 28 hours. The weekly variance σ²_weekly = 7 * (1)² = 7. Therefore, the standard deviation σ_weekly = sqrt(7) ≈ 2.6458 hours.So, the weekly gaming hours follow a normal distribution with μ=28 and σ≈2.6458.The friend agrees to join if the total time is within one standard deviation above or below the average. So, that would be the range [μ - σ, μ + σ].Calculating that: 28 - 2.6458 ≈ 25.3542 and 28 + 2.6458 ≈ 30.6458.Therefore, the friend agrees to join if the total gaming hours are between approximately 25.35 and 30.65 hours per week.Wait, but let me think again. The problem says the friend's availability follows a normal distribution similar to the gamer's daily gaming hours but scaled up to weekly hours. So, does that mean the friend's weekly hours also have the same distribution as the sum of seven daily hours? Or is it that the friend's daily hours are similar to the gamer's, but scaled up?Wait, the wording is a bit ambiguous. It says, "the friend's availability follows a normal distribution similar to the gamer's daily gaming hours but scaled up to weekly hours." So, perhaps the friend's daily hours are similar (same μ and σ) but scaled up to weekly, meaning the friend's weekly hours are also a sum of seven daily hours, just like the gamer's. So, same distribution as the gamer's weekly hours.But the problem says the friend will agree only if the total time spent gaming together in a week is within one standard deviation above or below the average weekly gaming hours of the dedicated gamer. So, the average weekly gaming hours of the dedicated gamer is 28, with a standard deviation of sqrt(7). So, the range is 28 ± sqrt(7), which is approximately 28 ± 2.6458, so 25.3542 to 30.6458.Therefore, the friend agrees if the total gaming hours are between roughly 25.35 and 30.65 hours.Wait, but hold on. Is the total time spent gaming together in a week the sum of both the gamer's and the friend's hours? Or is it just the gamer's hours? The question says, "the total time spent gaming together in a week is within one standard deviation above or below the average weekly gaming hours of the dedicated gamer."Hmm, so it's the total time spent gaming together, which would be the sum of both the gamer's and the friend's weekly hours. But the average weekly gaming hours of the dedicated gamer is 28. So, does that mean the total time (gamer + friend) should be within one standard deviation of 28? Or is it that the total time should be within one standard deviation of the average, which is 28, but considering the combined distribution?Wait, the problem says, "the total time spent gaming together in a week is within one standard deviation above or below the average weekly gaming hours of the dedicated gamer." So, the average is 28, and the standard deviation is sqrt(7). So, the range is 28 ± sqrt(7). So, regardless of the friend's hours, as long as the total is within that range, the friend will join.But wait, the friend's availability follows a normal distribution similar to the gamer's daily gaming hours but scaled up to weekly hours. So, the friend's weekly hours are also a normal distribution with μ=28 and σ=sqrt(7). So, if both are gaming together, the total time would be the sum of two independent normal variables, each with μ=28 and σ=sqrt(7). Therefore, the total would have μ_total = 28 + 28 = 56, and σ_total = sqrt(7 + 7) = sqrt(14) ≈ 3.7417.But the problem says the friend agrees if the total time is within one standard deviation of the dedicated gamer's average weekly hours, which is 28. So, the total time should be within [28 - sqrt(7), 28 + sqrt(7)] ≈ [25.3542, 30.6458].But wait, if the total time is the sum of both gamers, which is 56 on average, how can it be within 25 to 30? That doesn't make sense because 25 to 30 is much lower than the average total of 56. So, perhaps I misinterpreted the question.Let me read it again: "the friend agrees to join only if the total time spent gaming together in a week is within one standard deviation above or below the average weekly gaming hours of the dedicated gamer."Ah, maybe it's not the sum of both, but just the dedicated gamer's hours. So, the total time spent gaming together is just the dedicated gamer's hours, and the friend will join only if that total is within one standard deviation of the average. So, the friend is considering joining based on the dedicated gamer's weekly hours, not the combined total.Wait, that might make more sense. So, the friend is looking at the dedicated gamer's weekly hours and will join if that is within one standard deviation of the average. So, the dedicated gamer's weekly hours have μ=28 and σ=sqrt(7). So, the range is 28 ± sqrt(7), which is approximately 25.35 to 30.65.Therefore, the friend agrees to join if the dedicated gamer's weekly hours are within that range. So, the range is [28 - sqrt(7), 28 + sqrt(7)].Alternatively, if the total time is the sum of both, then the total would have μ=56 and σ=sqrt(14). But the friend is considering the total time relative to the dedicated gamer's average, which is 28. So, maybe it's not the sum but just the dedicated gamer's hours.I think the correct interpretation is that the total time spent gaming together is the dedicated gamer's weekly hours, and the friend will join if that total is within one standard deviation of the dedicated gamer's average. So, the range is [28 - sqrt(7), 28 + sqrt(7)].Therefore, the range is approximately 25.35 to 30.65 hours.Wait, but let me think again. If the friend is considering the total time spent gaming together, that would logically be the sum of both their hours. But the problem says it's within one standard deviation of the dedicated gamer's average weekly hours. So, the dedicated gamer's average is 28, so the total time should be within 28 ± sqrt(7). But if the total time is the sum, which is 56 on average, then 28 is much lower than that. So, that doesn't make sense.Alternatively, maybe the friend is considering the dedicated gamer's hours as the total time. So, if the dedicated gamer's weekly hours are within one standard deviation of their own average, then the friend will join. So, the friend is only interested in the dedicated gamer's consistency, not the combined time.That seems plausible. So, the friend will join if the dedicated gamer's weekly hours are within [28 - sqrt(7), 28 + sqrt(7)]. So, the range is approximately 25.35 to 30.65 hours.Yes, that makes sense. So, the friend is looking at the dedicated gamer's consistency, not the combined total. So, the range is based on the dedicated gamer's weekly hours.Therefore, the answer to the second question is that the friend agrees to join if the total gaming hours are between approximately 25.35 and 30.65 hours per week.Wait, but let me make sure. The problem says, "the total time spent gaming together in a week is within one standard deviation above or below the average weekly gaming hours of the dedicated gamer." So, "total time spent gaming together" could imply both of them. But if that's the case, as I thought earlier, the total would be the sum, which is 56 on average, and the standard deviation would be sqrt(14). So, one standard deviation would be 56 ± sqrt(14). But the problem says it's within one standard deviation of the dedicated gamer's average, which is 28. So, that seems conflicting.Wait, maybe the standard deviation is still based on the dedicated gamer's weekly hours, not the combined. So, the total time (which is the sum) should be within one standard deviation of the dedicated gamer's average, which is 28. So, the range is 28 ± sqrt(7). But the total time is 56 on average, so 28 is much lower. That doesn't make sense because the total time can't be less than the individual's time.Wait, perhaps the problem is not considering the sum but just the dedicated gamer's time. So, the total time spent gaming together is just the dedicated gamer's time, and the friend will join if that is within one standard deviation of the average. So, the range is 28 ± sqrt(7). That seems more plausible.Alternatively, maybe the friend's availability is such that their weekly hours are similar to the gamer's daily hours scaled up. So, the friend's weekly hours are also normal with μ=28 and σ=sqrt(7). So, the total time together would be the sum, which is 56 with σ=sqrt(14). But the friend is only interested in the total time being within one standard deviation of the dedicated gamer's average, which is 28. So, the total time should be within 28 ± sqrt(7). But that would mean the total time is between 25.35 and 30.65, which is much lower than the expected total of 56. That doesn't make sense because the total time can't be less than the individual's time.Therefore, perhaps the correct interpretation is that the friend is considering the dedicated gamer's weekly hours as the total time, and will join if that is within one standard deviation of the average. So, the range is [28 - sqrt(7), 28 + sqrt(7)].Yes, that seems to be the only logical conclusion. Otherwise, the numbers don't add up. So, the friend agrees if the dedicated gamer's weekly hours are within that range.Therefore, the range is approximately 25.35 to 30.65 hours.Wait, but let me think about the scaling again. The problem says the friend's availability follows a normal distribution similar to the gamer's daily gaming hours but scaled up to weekly hours. So, the daily hours for the friend are similar to the gamer's, which is μ=4, σ=1. So, the friend's weekly hours would be the sum of seven daily hours, same as the gamer's. Therefore, the friend's weekly hours are also N(28, sqrt(7)).But the problem says the friend will agree to join if the total time spent gaming together is within one standard deviation of the dedicated gamer's average weekly hours. So, the total time is the sum of both their weekly hours, which would be N(56, sqrt(14)). But the condition is that this total time is within one standard deviation of the dedicated gamer's average, which is 28. So, the range is 28 ± sqrt(7). But the total time is 56 on average, so 28 is much lower. That doesn't make sense because the total time can't be less than the individual's time.Wait, perhaps the problem is not considering the sum but just the dedicated gamer's time. So, the total time is just the dedicated gamer's time, and the friend will join if that is within one standard deviation of the average. So, the range is 28 ± sqrt(7). That seems to be the only way this makes sense.Therefore, the friend agrees to join if the dedicated gamer's weekly hours are between approximately 25.35 and 30.65 hours.Yes, that must be it. So, the answer is that range.So, summarizing:1. The probability that the gamer spends more than 5 hours gaming on a given day is approximately 15.87%.2. The friend agrees to join if the total gaming hours are between approximately 25.35 and 30.65 hours per week.Wait, but let me just confirm the scaling again. The friend's availability is similar to the gamer's daily hours but scaled up to weekly. So, if the gamer's daily is N(4,1), then the friend's daily is also N(4,1), so weekly is N(28, sqrt(7)). Therefore, the friend's weekly hours are the same distribution as the gamer's. So, if the total time is the sum, it's N(56, sqrt(14)). But the condition is based on the dedicated gamer's average, which is 28. So, the total time should be within 28 ± sqrt(7). But that would mean the total time is less than the dedicated gamer's average, which is impossible because the total is the sum of both.Therefore, the only logical conclusion is that the total time is just the dedicated gamer's time, and the friend is considering whether that is within one standard deviation of the average. So, the range is [28 - sqrt(7), 28 + sqrt(7)].Yes, that must be it. So, the friend is only concerned with the dedicated gamer's consistency, not the combined total. Therefore, the range is approximately 25.35 to 30.65 hours.Alright, I think I've thought this through enough. Time to write the final answers.</think>

🧠:<think>Okay, so I have this problem about a book club for cancer survivors, and I need to figure out two things: the total number of participants over 12 months and the month(s) where the product of participants and books discussed is the highest. Let me try to break this down step by step.First, let's tackle the first question: calculating the total number of participants over the 12-month period. The number of participants each month is given by the sequence ( P_n = 30 + 5n - n^2 ), where ( n ) is the month number from 1 to 12. So, I need to compute ( P_n ) for each month and then sum them all up.Hmm, okay. Let me write down the formula again: ( P_n = 30 + 5n - n^2 ). So, for each month, I plug in the value of ( n ) and calculate the number of participants. Then, I'll add all those numbers together.Wait, but before I start calculating each month individually, maybe there's a smarter way. Since it's a quadratic sequence, perhaps I can find a formula for the sum of ( P_n ) from ( n = 1 ) to ( n = 12 ). Let me see.The general formula for the sum of a quadratic sequence ( an^2 + bn + c ) from ( n = 1 ) to ( N ) is:[sum_{n=1}^{N} (an^2 + bn + c) = a cdot frac{N(N+1)(2N+1)}{6} + b cdot frac{N(N+1)}{2} + c cdot N]In our case, the sequence is ( P_n = -n^2 + 5n + 30 ). So, ( a = -1 ), ( b = 5 ), and ( c = 30 ). Plugging these into the formula, we get:[sum_{n=1}^{12} P_n = (-1) cdot frac{12 cdot 13 cdot 25}{6} + 5 cdot frac{12 cdot 13}{2} + 30 cdot 12]Let me compute each term step by step.First term: ( (-1) cdot frac{12 cdot 13 cdot 25}{6} )Compute the numerator: 12 * 13 = 156; 156 * 25 = 3900So, 3900 divided by 6 is 650. Then, multiplied by -1: -650.Second term: ( 5 cdot frac{12 cdot 13}{2} )Compute 12 * 13 = 156; 156 / 2 = 78; 78 * 5 = 390.Third term: 30 * 12 = 360.Now, sum all three terms: -650 + 390 + 360.Let me compute that: -650 + 390 is -260; -260 + 360 is 100.Wait, so the total number of participants over 12 months is 100? Hmm, that seems low. Let me verify by calculating each month individually.Wait, maybe I made a mistake in the formula. Let me double-check the formula for the sum of squares and the sum of linear terms.The sum of squares formula is ( sum_{n=1}^{N} n^2 = frac{N(N+1)(2N+1)}{6} ), which is correct. The sum of linear terms is ( sum_{n=1}^{N} n = frac{N(N+1)}{2} ), which is also correct. And the sum of constants is ( c cdot N ).So, plugging in N=12:First term: ( (-1) cdot frac{12 cdot 13 cdot 25}{6} )Wait, 2N+1 when N=12 is 25, so that's correct. 12*13*25 is 3900, divided by 6 is 650, times -1 is -650.Second term: 5 * (12*13)/2 = 5 * 78 = 390.Third term: 30*12 = 360.Adding them: -650 + 390 is -260; -260 + 360 is 100. Hmm, so according to the formula, the total is 100. But let me compute each month's participants and add them up to see if that's correct.Let me list each month from n=1 to n=12:n=1: P1 = 30 + 5*1 - 1^2 = 30 + 5 - 1 = 34n=2: P2 = 30 + 10 - 4 = 36n=3: 30 + 15 - 9 = 36n=4: 30 + 20 - 16 = 34n=5: 30 + 25 - 25 = 30n=6: 30 + 30 - 36 = 24n=7: 30 + 35 - 49 = 16n=8: 30 + 40 - 64 = -24? Wait, that can't be right. Number of participants can't be negative.Wait, hold on. Maybe I made a mistake in the formula? Let me check n=8 again.Wait, P8 = 30 + 5*8 - 8^2 = 30 + 40 - 64 = 70 - 64 = 6. Oh, I must have miscalculated earlier.Wait, 30 + 40 is 70, minus 64 is 6. So, P8=6.Wait, so n=8: 6 participants.Similarly, let's compute all months:n=1: 30 +5 -1=34n=2:30+10-4=36n=3:30+15-9=36n=4:30+20-16=34n=5:30+25-25=30n=6:30+30-36=24n=7:30+35-49=16n=8:30+40-64=6n=9:30+45-81= -6? Wait, that can't be. 30+45=75-81=-6? That's negative. Hmm, participants can't be negative. Maybe the formula is only valid up to a certain point?Wait, perhaps the formula is designed such that participants don't go negative, but in reality, the number of participants can't be negative, so maybe after a certain month, the number of participants is zero or something.Wait, let me compute n=9: 30 + 5*9 - 9^2 = 30 +45 -81= -6. Hmm, negative. So maybe the formula is incorrect beyond a certain point, or perhaps the book club stops when participants become zero or negative.Wait, but the problem says it's over a span of 12 months, so maybe the formula is correct, but participants can't be negative, so in reality, participants would be zero from n=9 onwards.But the problem didn't specify that, so perhaps we just take the formula as is, even if it gives negative numbers. So, P9=-6, P10=30+50-100= -20, P11=30+55-121= -36, P12=30+60-144= -54.Wait, that would mean participants are negative, which doesn't make sense. So, perhaps the formula is only valid up to a certain month where Pn becomes zero or positive.Wait, let me check when Pn becomes negative.Set Pn = 30 +5n -n² >=0So, -n² +5n +30 >=0Multiply both sides by -1 (inequality flips):n² -5n -30 <=0Find the roots of n² -5n -30=0Using quadratic formula: n=(5 ± sqrt(25 +120))/2 = (5 ± sqrt(145))/2 ≈ (5 ±12.0416)/2So, positive root is (5 +12.0416)/2≈17.0416/2≈8.5208So, the inequality n² -5n -30 <=0 holds for n between the two roots, which are approximately -7.0416 and 8.5208. Since n is positive, the inequality holds for n <=8.5208. So, for n=1 to 8, Pn is positive, and for n=9 to 12, Pn is negative.But participants can't be negative, so perhaps in reality, participants are zero for n=9 to 12. So, maybe the formula is only valid up to n=8, and beyond that, participants are zero.But the problem didn't specify that, so perhaps we should take the formula as given, even if it results in negative participants. But that doesn't make sense in real life.Alternatively, maybe the formula is correct, but participants can't be negative, so for n where Pn is negative, participants are zero.So, perhaps for n=9 to 12, Pn=0.But the problem didn't specify, so maybe we should just compute as per the formula, even if it's negative.But that would lead to negative participants, which is impossible. So, perhaps the formula is only valid up to n=8, and beyond that, participants are zero.Alternatively, maybe the formula is correct, and the negative participants are just an artifact, and we should take the absolute value or something. But that also doesn't make sense.Wait, maybe I made a mistake in the formula. Let me check again.Wait, the formula is Pn=30 +5n -n². So, for n=1: 30+5-1=34, correct. n=2:30+10-4=36, correct. n=3:30+15-9=36, correct. n=4:30+20-16=34, correct. n=5:30+25-25=30, correct. n=6:30+30-36=24, correct. n=7:30+35-49=16, correct. n=8:30+40-64=6, correct. n=9:30+45-81=-6, which is negative. So, perhaps the formula is correct, but participants can't be negative, so for n=9 to 12, participants are zero.So, in that case, the total participants would be sum from n=1 to 8 of Pn, plus zero for n=9 to 12.So, let's compute the sum from n=1 to 8:n=1:34n=2:36n=3:36n=4:34n=5:30n=6:24n=7:16n=8:6Let me add these up:34 +36=7070+36=106106+34=140140+30=170170+24=194194+16=210210+6=216So, total participants from n=1 to 8 is 216. Then, n=9 to 12: 0 each, so total is 216.But earlier, using the formula, I got 100, which is way off. So, clearly, the formula approach was incorrect because it included negative participants.So, the correct approach is to compute each month's participants and sum them up, considering that participants can't be negative. So, for n=1 to 8, participants are as calculated, and for n=9 to 12, participants are zero.So, total participants: 216.Wait, but let me double-check the sum:n=1:34n=2:36 (34+36=70)n=3:36 (70+36=106)n=4:34 (106+34=140)n=5:30 (140+30=170)n=6:24 (170+24=194)n=7:16 (194+16=210)n=8:6 (210+6=216)Yes, that's correct.So, the total number of participants over 12 months is 216.Wait, but the formula gave me 100, which is way off. So, the formula approach was incorrect because it included negative participants, which we have to disregard.Therefore, the correct total is 216.Okay, so that's the first part.Now, moving on to the second question: Determine the month(s) in which the product of the number of participants and the number of books discussed is maximized.So, we need to compute the product ( P_n times B_n ) for each month n from 1 to 12, and find the month(s) where this product is the highest.Given that ( P_n = 30 +5n -n^2 ) and ( B_n = 2 + lfloor sqrt{n} rfloor ).First, let's understand ( B_n ). The floor function of sqrt(n) means we take the greatest integer less than or equal to sqrt(n). So, for each n from 1 to 12, let's compute ( B_n ).Let me list n from 1 to 12 and compute ( sqrt{n} ) and then floor it:n=1: sqrt(1)=1.0, floor=1, so B1=2+1=3n=2: sqrt(2)≈1.414, floor=1, B2=3n=3: sqrt(3)≈1.732, floor=1, B3=3n=4: sqrt(4)=2.0, floor=2, B4=2+2=4n=5: sqrt(5)≈2.236, floor=2, B5=4n=6: sqrt(6)≈2.449, floor=2, B6=4n=7: sqrt(7)≈2.645, floor=2, B7=4n=8: sqrt(8)≈2.828, floor=2, B8=4n=9: sqrt(9)=3.0, floor=3, B9=2+3=5n=10: sqrt(10)≈3.162, floor=3, B10=5n=11: sqrt(11)≈3.316, floor=3, B11=5n=12: sqrt(12)≈3.464, floor=3, B12=5So, ( B_n ) is 3 for n=1-3, 4 for n=4-8, and 5 for n=9-12.Now, let's compute the product ( P_n times B_n ) for each month.But remember, for n=9-12, ( P_n ) is negative, but participants can't be negative, so we should consider ( P_n ) as zero for n=9-12.Wait, but the problem didn't specify that, so perhaps we should use the formula as given, even if participants are negative. But in reality, participants can't be negative, so the product would be negative, which doesn't make sense in this context.Therefore, perhaps for n=9-12, ( P_n ) is zero, so the product is zero.Alternatively, if we take the formula as is, the product could be negative, but since we're looking for the maximum product, negative products would be less than positive ones, so the maximum would occur before n=9.But let's proceed step by step.First, let's compute ( P_n ) and ( B_n ) for each month, then compute the product.n=1:P1=34, B1=3, product=34*3=102n=2:P2=36, B2=3, product=36*3=108n=3:P3=36, B3=3, product=36*3=108n=4:P4=34, B4=4, product=34*4=136n=5:P5=30, B5=4, product=30*4=120n=6:P6=24, B6=4, product=24*4=96n=7:P7=16, B7=4, product=16*4=64n=8:P8=6, B8=4, product=6*4=24n=9:P9=-6, B9=5, product=-6*5=-30n=10:P10=-20, B10=5, product=-20*5=-100n=11:P11=-36, B11=5, product=-36*5=-180n=12:P12=-54, B12=5, product=-54*5=-270Now, let's list the products:n=1:102n=2:108n=3:108n=4:136n=5:120n=6:96n=7:64n=8:24n=9:-30n=10:-100n=11:-180n=12:-270So, looking at these products, the highest product is 136 at n=4.Wait, but let me double-check the calculations for each month to make sure I didn't make any errors.n=1:34*3=102, correct.n=2:36*3=108, correct.n=3:36*3=108, correct.n=4:34*4=136, correct.n=5:30*4=120, correct.n=6:24*4=96, correct.n=7:16*4=64, correct.n=8:6*4=24, correct.n=9:-6*5=-30, correct.n=10:-20*5=-100, correct.n=11:-36*5=-180, correct.n=12:-54*5=-270, correct.So, the maximum product is indeed 136, occurring in month n=4, which is April.Wait, but let me check if there are any other months with the same product. For example, n=2 and n=3 both have 108, which is less than 136. So, the maximum is only in April.Therefore, the month(s) where the product is maximized is April (n=4).Wait, but let me think again. The product is 136 in April, which is higher than all other months. So, that's the maximum.Therefore, the answers are:1. Total participants: 2162. The product is maximized in April (n=4).But wait, just to make sure, let me compute the product for n=4 again: P4=34, B4=4, 34*4=136. Correct.And for n=5: P5=30, B5=4, 30*4=120, which is less.Yes, so April is the month with the highest product.So, summarizing:1. Total participants over 12 months: 2162. The product is maximized in April (month 4).Final Answer1. The total number of participants over the 12-month period is boxed{216}.2. The product of participants and books discussed is maximized in month boxed{4}.</think>

🧠:<think>Alright, so I have this problem about a grandchild trying to understand how a healthy diet can affect their grandparents' lifespan. The problem is divided into two sub-problems. Let me tackle them one by one.Starting with Sub-problem 1: Determine the maximum life expectancy ( L(P) ) and the corresponding percentage ( P ) of a healthy diet that maximizes this life expectancy. The function given is:[ L(P) = 78 + 0.2P + 0.01P^2 ]Hmm, okay. So this is a quadratic function in terms of ( P ). Quadratic functions have the form ( ax^2 + bx + c ), and their graphs are parabolas. Since the coefficient of ( P^2 ) is positive (0.01), the parabola opens upwards, which means the vertex is the minimum point. Wait, that can't be right because the problem is asking for the maximum life expectancy. If the parabola opens upwards, there isn't a maximum—it goes to infinity as ( P ) increases. But that doesn't make sense in the context of the problem because ( P ) is a percentage and can only range from 0 to 100.Wait, maybe I made a mistake. Let me double-check. The function is ( L(P) = 78 + 0.2P + 0.01P^2 ). So, yes, the coefficient of ( P^2 ) is positive, meaning it's a convex function, opening upwards. Therefore, it doesn't have a maximum—it only has a minimum. But the problem is asking for the maximum life expectancy. That seems contradictory.Hold on, maybe I misread the function. Let me look again. It says ( L(P) = 78 + 0.2P + 0.01P^2 ). So, it's definitely a quadratic with a positive leading coefficient. Hmm. So, in that case, as ( P ) increases beyond a certain point, ( L(P) ) will keep increasing. But since ( P ) is a percentage, it can't go beyond 100. So, actually, the maximum life expectancy would occur at the maximum value of ( P ), which is 100.Wait, but if the parabola opens upwards, the minimum is at the vertex, and the maximum would be at the endpoints of the domain. Since ( P ) is between 0 and 100, the maximum ( L(P) ) would be at ( P = 100 ).But that seems a bit odd because usually, in these kinds of models, there might be an optimal point where life expectancy peaks and then maybe decreases. Maybe the function was supposed to be concave, meaning the coefficient of ( P^2 ) is negative? Let me check the original problem again.No, the function is definitely ( 78 + 0.2P + 0.01P^2 ). So, it's a convex function. Hmm. So, in this case, the life expectancy increases as ( P ) increases, with the rate of increase accelerating because of the quadratic term. So, the maximum life expectancy would be at ( P = 100 ).But let me think again. Maybe the grandchild made a mistake in modeling. If it's a quadratic function with a positive coefficient on ( P^2 ), it's convex, so it doesn't have a maximum—it only has a minimum. So, in the context of ( P ) from 0 to 100, the maximum life expectancy would be at ( P = 100 ).Wait, but if I calculate the derivative, maybe I can find a critical point. Let me try that.Taking the derivative of ( L(P) ) with respect to ( P ):[ L'(P) = 0.2 + 0.02P ]Setting the derivative equal to zero to find critical points:[ 0.2 + 0.02P = 0 ][ 0.02P = -0.2 ][ P = -10 ]Hmm, so the critical point is at ( P = -10 ), which is outside the domain of ( P ) (0 to 100). So, within the domain, the function is increasing because the derivative is positive for all ( P > -10 ). Since ( P ) is always positive, the derivative is always positive, meaning the function is increasing on the interval [0, 100]. Therefore, the maximum life expectancy occurs at ( P = 100 ).So, plugging ( P = 100 ) into the function:[ L(100) = 78 + 0.2(100) + 0.01(100)^2 ][ L(100) = 78 + 20 + 100 ][ L(100) = 198 ]Wait, that seems extremely high. The average lifespan is 78 years, and with a 100% healthy diet, it's 198 years? That doesn't seem realistic. Maybe the model is flawed, or perhaps the coefficients are incorrect. But according to the problem, that's the function given.Alternatively, maybe the function was supposed to be concave, with a negative coefficient on ( P^2 ). If that were the case, the parabola would open downward, and there would be a maximum at the vertex. But as per the problem statement, it's positive.So, unless there's a typo, I have to go with the given function. Therefore, the maximum life expectancy is 198 years at ( P = 100 ). But that seems unrealistic. Maybe the grandchild's model is incorrect, but since we have to work with it, I'll proceed.Wait, perhaps I misread the function. Let me check again. It says:[ L(P) = 78 + 0.2P + 0.01P^2 ]Yes, that's correct. So, with that, the maximum is at ( P = 100 ), giving ( L = 198 ). But that's a huge increase. Maybe the units are different? Or perhaps it's a misprint, and it should be ( -0.01P^2 ). But without that information, I can't assume.Alternatively, maybe the function is meant to be a quadratic that peaks somewhere within 0 to 100. Let me consider that possibility. If the function were concave, the maximum would be at the vertex. But since it's convex, the maximum is at the upper bound.So, perhaps the answer is that the maximum life expectancy is 198 years when ( P = 100 ). But that seems too high. Maybe the grandchild's model is incorrect, but I have to go with the given function.Wait, maybe I made a mistake in calculating ( L(100) ). Let me recalculate:[ L(100) = 78 + 0.2*100 + 0.01*(100)^2 ][ = 78 + 20 + 100 ][ = 198 ]Yes, that's correct. So, according to the model, a 100% healthy diet would result in a life expectancy of 198 years. That's a 120-year increase, which is not feasible in reality. So, perhaps the model is incorrect, but since we have to use it, I'll proceed.Therefore, for Sub-problem 1, the maximum life expectancy is 198 years at ( P = 100 ).Wait, but let me think again. Maybe the function is supposed to be a concave function, so the maximum is at the vertex. Let me assume for a moment that the coefficient of ( P^2 ) is negative. If it were ( -0.01P^2 ), then the function would be concave, and the maximum would be at the vertex.Let me recast the function as:[ L(P) = 78 + 0.2P - 0.01P^2 ]Then, the derivative would be:[ L'(P) = 0.2 - 0.02P ]Setting derivative to zero:[ 0.2 - 0.02P = 0 ][ 0.02P = 0.2 ][ P = 10 ]So, the maximum would be at ( P = 10 ), and ( L(10) = 78 + 2 - 1 = 79 ). That seems more reasonable, but the original function is given with a positive coefficient. So, unless there's a typo, I have to go with the original.But given that the result is unrealistic, maybe I should consider that perhaps the function is supposed to be concave. Alternatively, perhaps the function is correct, and the model is just an approximation that doesn't account for diminishing returns or other factors beyond a certain point.Alternatively, maybe the function is correct, and the maximum is indeed at ( P = 100 ), giving 198 years. But that seems too high. Maybe the coefficients are in different units. Let me check the units again.The function is given as:[ L(P) = 78 + 0.2P + 0.01P^2 ]Where ( P ) is the percentage (0 to 100). So, 0.2P would be in years, and 0.01P^2 would also be in years. So, for ( P = 100 ), 0.2*100 = 20, and 0.01*100^2 = 100. So, total is 78 + 20 + 100 = 198. So, the units check out, but the result is unrealistic.Alternatively, maybe the function is supposed to be in terms of years added, not total years. But the problem states it's the life expectancy, so it's total years.Given that, perhaps the function is correct, and the maximum is indeed 198 years at 100% healthy diet. So, I'll proceed with that.Now, moving on to Sub-problem 2: If the grandchild wants their grandparents to achieve at least an 85-year life expectancy, within what range of ( P ) should their grandparents' healthy diet percentage be?So, we need to find the values of ( P ) such that ( L(P) geq 85 ).Given:[ 78 + 0.2P + 0.01P^2 geq 85 ]Subtracting 85 from both sides:[ 0.01P^2 + 0.2P + 78 - 85 geq 0 ][ 0.01P^2 + 0.2P - 7 geq 0 ]So, we have a quadratic inequality:[ 0.01P^2 + 0.2P - 7 geq 0 ]Let me write it as:[ 0.01P^2 + 0.2P - 7 geq 0 ]To solve this inequality, first, let's find the roots of the equation:[ 0.01P^2 + 0.2P - 7 = 0 ]Multiply both sides by 100 to eliminate decimals:[ P^2 + 20P - 700 = 0 ]Now, using the quadratic formula:[ P = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]Where ( a = 1 ), ( b = 20 ), ( c = -700 ).Calculating the discriminant:[ D = 20^2 - 4*1*(-700) ][ D = 400 + 2800 ][ D = 3200 ]So,[ P = frac{-20 pm sqrt{3200}}{2} ][ sqrt{3200} = sqrt{100*32} = 10*5.6568 ≈ 56.568 ]So,[ P = frac{-20 + 56.568}{2} ≈ frac{36.568}{2} ≈ 18.284 ][ P = frac{-20 - 56.568}{2} ≈ frac{-76.568}{2} ≈ -38.284 ]Since ( P ) can't be negative, we discard the negative root. So, the critical point is at approximately ( P ≈ 18.284 ).Now, since the quadratic opens upwards (because the coefficient of ( P^2 ) is positive), the inequality ( 0.01P^2 + 0.2P - 7 geq 0 ) holds when ( P leq -38.284 ) or ( P geq 18.284 ). But since ( P ) can't be negative, the solution is ( P geq 18.284 ).Therefore, the grandparents need to have at least approximately 18.284% of their diet as healthy to achieve a life expectancy of at least 85 years.But since ( P ) is a percentage, we can round it to two decimal places or to a whole number. Let me calculate it more precisely.The exact value of ( sqrt{3200} ) is ( sqrt{100*32} = 10sqrt{32} ). ( sqrt{32} = 4sqrt{2} ≈ 5.656854249 ). So,[ sqrt{3200} ≈ 10*5.656854249 ≈ 56.56854249 ]So,[ P = frac{-20 + 56.56854249}{2} ≈ frac{36.56854249}{2} ≈ 18.284271245 ]So, approximately 18.2843%.Therefore, the range of ( P ) is ( P geq 18.2843 ). Since ( P ) can't exceed 100, the range is ( 18.2843 leq P leq 100 ).But let me verify this by plugging ( P = 18.2843 ) into the original function:[ L(18.2843) = 78 + 0.2*18.2843 + 0.01*(18.2843)^2 ]First, calculate each term:0.2*18.2843 ≈ 3.656860.01*(18.2843)^2 ≈ 0.01*(334.333) ≈ 3.34333Adding up:78 + 3.65686 + 3.34333 ≈ 78 + 7 ≈ 85Yes, that checks out. So, at ( P ≈ 18.2843 ), ( L(P) ≈ 85 ).Therefore, to achieve at least an 85-year life expectancy, the grandparents need to have at least approximately 18.28% of their diet as healthy. So, the range is ( P geq 18.28 ).But since percentages are often given to one decimal place, we can say ( P geq 18.3 ).Alternatively, if we want to express it as a range, it's from approximately 18.3% to 100%.Wait, but let me think again. The quadratic inequality is ( 0.01P^2 + 0.2P - 7 geq 0 ). Since the parabola opens upwards, the inequality is satisfied when ( P leq ) the smaller root or ( P geq ) the larger root. But since the smaller root is negative, we only consider ( P geq 18.2843 ).Therefore, the range is ( P geq 18.2843 ), which is approximately 18.28%.So, summarizing:Sub-problem 1: The maximum life expectancy is 198 years at ( P = 100 ).Sub-problem 2: The grandparents need to have at least approximately 18.28% of their diet as healthy to achieve a life expectancy of at least 85 years.But wait, let me cross-verify the calculations for Sub-problem 2. I want to make sure I didn't make any errors.Starting from the inequality:[ 0.01P^2 + 0.2P - 7 geq 0 ]Multiplying by 100:[ P^2 + 20P - 700 geq 0 ]Quadratic formula:[ P = frac{-20 pm sqrt{400 + 2800}}{2} ][ P = frac{-20 pm sqrt{3200}}{2} ][ sqrt{3200} = 56.5685 ]So,[ P = frac{-20 + 56.5685}{2} ≈ 18.2843 ][ P = frac{-20 - 56.5685}{2} ≈ -38.2843 ]Yes, that's correct. So, the solution is ( P geq 18.2843 ).Therefore, the range is ( P geq 18.2843 ), which is approximately 18.28%.So, the grandparents need to have at least about 18.28% of their diet as healthy to reach an 85-year life expectancy.But let me check what happens if ( P = 18 ):[ L(18) = 78 + 0.2*18 + 0.01*(18)^2 ][ = 78 + 3.6 + 3.24 ][ = 84.84 ]Which is just below 85. So, 18% gives 84.84 years, which is less than 85. Therefore, they need a bit more than 18%.At ( P = 18.2843 ), we get exactly 85 years. So, the minimum ( P ) is approximately 18.28%.Therefore, the range is ( P geq 18.28% ).So, summarizing both sub-problems:Sub-problem 1: The maximum life expectancy is 198 years when ( P = 100% ).Sub-problem 2: To achieve at least an 85-year life expectancy, the grandparents need to have at least approximately 18.28% of their diet as healthy.But wait, in Sub-problem 1, the maximum life expectancy is at ( P = 100 ), which gives 198 years. That seems extremely high, but according to the model, that's the case. So, unless there's a mistake in the model, that's the answer.Alternatively, if the model were concave, the maximum would be at the vertex, but since it's convex, the maximum is at the upper bound.Therefore, I think that's the correct approach.</think>