🧠:<think>Alright, so I have these two statistics problems to solve, and I need to figure them out step by step. Let me take them one at a time.Starting with problem 1: Dr. Amina has data from two countries, X and Y, each with a sample size of 100 individuals. The mean leisure hours for Country X is 15 hours with a standard deviation of 4, and for Country Y, it's 12 hours with a standard deviation of 5. She wants a 95% confidence interval for the mean difference in leisure hours between the two countries. Okay, so confidence intervals for the difference in means. Since the samples are from two independent countries, we're dealing with two independent samples. The formula for the confidence interval is (mean1 - mean2) ± (critical value) * (standard error). First, I need to find the difference in means. That's 15 - 12, which is 3 hours. So the point estimate is 3 hours. Next, the standard error (SE) for the difference in means. The formula for SE is sqrt[(s1^2/n1) + (s2^2/n2)]. Plugging in the numbers: s1 is 4, n1 is 100; s2 is 5, n2 is 100. So, s1 squared is 16, divided by 100 is 0.16. s2 squared is 25, divided by 100 is 0.25. Adding them together gives 0.41. Taking the square root of 0.41, which is approximately 0.6403. Now, the critical value for a 95% confidence interval. Since the sample sizes are large (100 each), we can use the z-score. For 95% confidence, the z-score is 1.96. So, the margin of error is 1.96 * 0.6403. Let me calculate that: 1.96 * 0.6403 is approximately 1.254. Therefore, the confidence interval is 3 ± 1.254, which gives us a lower bound of 1.746 and an upper bound of 4.254. So, we can be 95% confident that the true mean difference in leisure hours between Country X and Country Y is between approximately 1.75 and 4.25 hours.Moving on to problem 2: Dr. Amina wants to perform a two-sample t-test to test the null hypothesis that there's no difference in average leisure hours between the two countries at a 0.05 significance level.Alright, so setting up the hypothesis test. The null hypothesis (H0) is that the mean difference is zero, and the alternative hypothesis (H1) is that the mean difference is not zero (two-tailed test).Given that both sample sizes are large (n=100), we can use the z-test instead of the t-test, but since the question mentions a t-test, I think it's expecting us to use the t-test formula. However, with such large sample sizes, the t-test and z-test will give very similar results.The test statistic for a two-sample t-test is (mean1 - mean2) / SE, where SE is the same as before, sqrt[(s1^2/n1) + (s2^2/n2)]. We already calculated that as approximately 0.6403.So, the test statistic is 3 / 0.6403, which is approximately 4.685. Now, determining the degrees of freedom. For a two-sample t-test, the degrees of freedom can be calculated using the Welch-Satterthwaite equation, which is [(s1^2/n1 + s2^2/n2)^2] / [(s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1)]. Plugging in the numbers:Numerator: (0.16 + 0.25)^2 = (0.41)^2 = 0.1681Denominator: (0.16^2)/99 + (0.25^2)/99 = (0.0256 + 0.0625)/99 = 0.0881/99 ≈ 0.00089So, degrees of freedom ≈ 0.1681 / 0.00089 ≈ 189. Since the degrees of freedom are large, the t-distribution is very close to the z-distribution. The critical t-value for a two-tailed test at 0.05 significance level with 189 degrees of freedom is approximately ±1.97. Our test statistic is 4.685, which is much larger than 1.97. Therefore, we reject the null hypothesis. Alternatively, if we calculate the p-value, which is the probability of observing a test statistic as extreme as 4.685 under the null hypothesis. Given that the t-distribution with 189 degrees of freedom is almost normal, the p-value would be extremely small, much less than 0.05. So, we have sufficient evidence to conclude that there is a statistically significant difference in the average leisure hours between Country X and Country Y at the 0.05 significance level.Wait, but hold on. Since both sample sizes are 100, which is quite large, the Central Limit Theorem tells us that the sampling distribution of the difference in means will be approximately normal, so using a z-test would also be appropriate here. The z-test would give the same conclusion because the z-score is 4.685, which is way beyond the critical value of 1.96. So, either way, the conclusion is the same.Just to recap, the steps were:1. Calculate the difference in sample means.2. Compute the standard error of the difference.3. Determine the critical value based on the significance level and degrees of freedom.4. Calculate the test statistic.5. Compare the test statistic to the critical value or find the p-value.6. Make a decision to reject or fail to reject the null hypothesis.Everything seems to check out. The confidence interval doesn't include zero, which also supports rejecting the null hypothesis. So, both the confidence interval and the hypothesis test lead to the same conclusion: there's a significant difference in leisure hours between the two countries.I think that's it. I don't see any mistakes in my calculations, but let me just double-check the standard error:s1^2/n1 = 16/100 = 0.16s2^2/n2 = 25/100 = 0.25Sum is 0.41, square root is ~0.6403. Correct.Test statistic: 3 / 0.6403 ≈ 4.685. Correct.Degrees of freedom calculation: Yes, using Welch-Satterthwaite, got approximately 189. So, t-critical is ~1.97. Our test statistic is much larger, so reject H0.Yes, all steps make sense. I think I'm confident with these answers.Final Answer1. The 95% confidence interval for the mean difference is boxed{(1.75, 4.25)} hours.2. The null hypothesis is rejected; there is a significant difference in average leisure hours between the two countries. The final answer is boxed{text{Reject } H_0}.</think>

🧠:<think>Okay, so I have this problem about an architect working on a multi-story building design. She's dealing with two sub-problems: one about the structural integrity of the building's framework and another about scheduling a video call considering time zones. Let me try to tackle each sub-problem step by step.Starting with Sub-problem 1: The building's framework is a 3D truss structure made of steel beams. Each beam has a length L, and the structure is modeled as a graph G with nodes as joints and edges as beams. The truss is subject to forces represented by matrix F, and the stiffness matrix K is such that Kx = F, where x is the displacement vector. We need to determine the conditions under which K is positive definite, ensuring the structure's stability.Hmm, positive definite matrices are important in structural engineering because they ensure the system is stable and doesn't have any mechanisms (unstable configurations). For a stiffness matrix K to be positive definite, a few conditions must be met.First, I remember that a symmetric matrix is positive definite if all its leading principal minors are positive. But in the context of structural mechanics, there's another way to think about it. The matrix K must be symmetric and positive definite, which relates to the structure being rigid and without any redundant constraints.Wait, in truss structures, positive definiteness of K implies that the structure is globally rigid. That is, there are no mechanisms or rigid motions that aren't accounted for by the supports. So, for K to be positive definite, the truss must be a minimally rigid graph, meaning it has just enough members to prevent any motion.But how does that translate into conditions on K? Well, K is a symmetric matrix, so that's one condition. For positive definiteness, all the eigenvalues of K must be positive. However, calculating eigenvalues for large matrices isn't practical, so maybe we can think in terms of the graph's properties.In graph theory, a truss structure is represented as a graph where nodes are joints and edges are beams. For the stiffness matrix to be positive definite, the graph must be rigid. In 2D, a rigid graph is one that is minimally rigid, meaning it has 2n - 3 edges for n nodes, and every subset of k nodes has at most 2k - 3 edges. But since this is a 3D truss, the conditions are a bit different.In 3D, a rigid graph requires 3n - 6 edges for n nodes, and every subset of k nodes has at most 3k - 6 edges. So, if the truss structure meets this condition, it should be rigid, and hence the stiffness matrix K should be positive definite. But wait, is that always the case?I think it's more nuanced. Even if the graph is rigid, the actual geometry and member orientations can affect the positive definiteness. For example, if the truss is over-constrained but in a way that introduces dependent constraints, it might still be positive definite. But in general, a minimally rigid graph in 3D with 3n - 6 edges should ensure that K is positive definite, provided there are no colinear or coplanar members that could cause rank deficiency.Also, the stiffness matrix K is typically assembled by combining the individual stiffness matrices of each beam. Each beam contributes to the global stiffness matrix based on its orientation and position. So, if the beams are properly connected and the structure doesn't have any internal mechanisms, K should be positive definite.Another thought: the structure must be properly supported. If the truss isn't adequately supported, it might not be rigid. So, the supports also play a role in the positive definiteness of K. If the supports are such that they prevent any rigid body motions, then K is positive definite.So, putting it all together, the conditions for K being positive definite are:1. The truss structure must be rigid, meaning it has sufficient members to prevent any motion. In 3D, this typically means the graph has 3n - 6 edges and satisfies the rigidity conditions for all subsets.2. The structure must be properly supported to prevent rigid body translations and rotations.3. The individual beams must be oriented such that their contributions to K don't cause any rank deficiency or negative eigenvalues.I think that covers the main points. Now, moving on to Sub-problem 2.The architect wants to schedule a video call with her team, who are in a time zone 8 hours behind her. She needs the call to be during working hours for both. Her available hours are 9 AM to 6 PM local time, and her team's working hours are the same in their local time.So, let's denote the architect's local time as T, and her team's time as T - 8 hours because they are 8 hours behind.We need to find the overlap between her working hours and her team's working hours.Her working hours: 9 AM to 6 PM T.Team's working hours: 9 AM to 6 PM (T - 8).So, let's convert the team's working hours into the architect's local time.Team's 9 AM is (9 AM + 8 hours) = 5 PM in architect's time.Team's 6 PM is (6 PM + 8 hours) = 2 AM next day in architect's time.Wait, that can't be right. If the team is 8 hours behind, then when it's 9 AM for the architect, it's 1 AM for the team. So, the team's working hours from 9 AM to 6 PM their time would be from 1 AM to 10 PM architect's time.Wait, I think I confused the direction. If the team is 8 hours behind, then when it's 9 AM for the architect, it's 1 AM for the team. So, the team's 9 AM is 1 AM architect's time, and their 6 PM is 10 PM architect's time.Therefore, the team's working hours in architect's local time are 1 AM to 10 PM.The architect's working hours are 9 AM to 6 PM.So, the overlap between 1 AM to 10 PM and 9 AM to 6 PM is 9 AM to 6 PM.Wait, but that would mean the entire architect's working hours overlap with the team's working hours. But that can't be right because the team's working hours in architect's time are 1 AM to 10 PM, which includes the architect's 9 AM to 6 PM.But the architect wants the call to be during her working hours and the team's working hours. So, the call must be scheduled when both are working. Since the team's working hours in architect's time are 1 AM to 10 PM, and the architect's are 9 AM to 6 PM, the overlap is 9 AM to 6 PM.But wait, that would mean the call can be scheduled anytime during the architect's working hours because the team is working from 1 AM to 10 PM, which includes the architect's 9 AM to 6 PM.But that seems too broad. Let me double-check.If the team is 8 hours behind, then when it's 9 AM for the architect, it's 1 AM for the team. The team's working hours are 9 AM to 6 PM their time, which is 1 PM to 11 PM architect's time.Wait, hold on, maybe I messed up the conversion.Let me think again. If the team is 8 hours behind, then their local time is T - 8.So, when it's 9 AM for the architect (T = 9 AM), it's 1 AM for the team (T - 8 = 1 AM).The team's working hours are 9 AM to 6 PM their time, which is 9 AM + 8 hours = 5 PM to 6 PM + 8 hours = 2 AM next day in architect's time.Wait, no. To convert the team's working hours to architect's time, we add 8 hours.So, team's 9 AM is 5 PM architect's time.Team's 6 PM is 2 AM next day architect's time.Therefore, the team's working hours in architect's time are 5 PM to 2 AM next day.But the architect's working hours are 9 AM to 6 PM.So, the overlap between 5 PM to 2 AM and 9 AM to 6 PM is 5 PM to 6 PM.Wait, that makes more sense. So, the overlapping time is from 5 PM to 6 PM architect's time.Therefore, the call must be scheduled between 5 PM and 6 PM in the architect's local time.But let me verify this again.Architect's time: 9 AM to 6 PM.Team's time: 9 AM to 6 PM, which is 8 hours behind.So, team's 9 AM is architect's 1 AM.Team's 6 PM is architect's 10 PM.So, team's working hours in architect's time: 1 AM to 10 PM.Architect's working hours: 9 AM to 6 PM.Overlap: 9 AM to 6 PM.Wait, now I'm confused because earlier I thought it was 5 PM to 6 PM, but now it's 9 AM to 6 PM.I think the confusion comes from whether we're adding or subtracting hours.If the team is 8 hours behind, then when it's 9 AM for the architect, it's 1 AM for the team.So, to find when the team is working (9 AM to 6 PM their time), we need to convert that to architect's time.Team's 9 AM = Architect's 1 AM.Team's 6 PM = Architect's 10 PM.So, the team's working hours in architect's time are 1 AM to 10 PM.Architect's working hours are 9 AM to 6 PM.So, the overlap is 9 AM to 6 PM.Therefore, the call can be scheduled anytime between 9 AM and 6 PM architect's time, because during that period, the team is working from 1 AM to 10 PM, which includes the architect's 9 AM to 6 PM.Wait, but that seems counterintuitive because the team is 8 hours behind. So, if the architect starts a call at 9 AM, it's 1 AM for the team, which is outside their working hours (they start at 9 AM their time, which is 1 PM architect's time).Wait, no. If the team's working hours are 9 AM to 6 PM their time, which is 1 PM to 10 PM architect's time.Therefore, the overlap between architect's 9 AM to 6 PM and team's 1 PM to 10 PM is 1 PM to 6 PM.So, the call must be scheduled between 1 PM and 6 PM architect's time.Wait, now I'm really confused. Let me try to make a table.Architect's time: 9 AM, 10 AM, 11 AM, 12 PM, 1 PM, 2 PM, 3 PM, 4 PM, 5 PM, 6 PM.Team's time: 1 AM, 2 AM, 3 AM, 4 AM, 5 AM, 6 AM, 7 AM, 8 AM, 9 AM, 10 AM.Wait, no. If the team is 8 hours behind, then when it's 9 AM for the architect, it's 1 AM for the team.So, the team's 9 AM is 1 PM architect's time.Team's 6 PM is 10 PM architect's time.Therefore, the team's working hours in architect's time are 1 PM to 10 PM.Architect's working hours: 9 AM to 6 PM.Overlap: 1 PM to 6 PM.So, the call must be scheduled between 1 PM and 6 PM architect's time.Therefore, the range of possible start times is from 1 PM to 6 PM.But wait, the architect's available time is 9 AM to 6 PM, and the team is available from 1 PM to 10 PM in architect's time.So, the overlap is 1 PM to 6 PM.Therefore, the call can start anytime between 1 PM and 6 PM in the architect's local time.But let me check with specific times.If the architect starts at 1 PM, it's 5 AM for the team, which is within their 9 AM to 6 PM? No, 5 AM is before their 9 AM.Wait, hold on. If the team is 8 hours behind, then when it's 1 PM for the architect, it's 5 AM for the team.But the team's working hours are 9 AM to 6 PM their time, which is 5 PM to 2 AM next day in architect's time.Wait, I think I'm mixing up the direction.Let me define:Architect's time (A) = Team's time (T) + 8 hours.So, T = A - 8.Team's working hours: 9 AM to 6 PM T.Convert to A: 9 AM T = 5 PM A, 6 PM T = 2 AM next day A.So, team's working hours in A's time: 5 PM to 2 AM next day.Architect's working hours: 9 AM to 6 PM A.Overlap between 5 PM to 2 AM and 9 AM to 6 PM is 5 PM to 6 PM.Therefore, the call must be scheduled between 5 PM and 6 PM A.Wait, that makes sense because from 5 PM to 6 PM A, it's 9 AM to 10 AM T, which is within the team's working hours.So, the overlap is only 1 hour: 5 PM to 6 PM A.Therefore, the range of possible start times is 5 PM to 6 PM.But earlier I thought it was 1 PM to 6 PM, but that was incorrect because I didn't properly convert the team's working hours.So, the correct overlap is 5 PM to 6 PM.Therefore, the architect should schedule the call between 5 PM and 6 PM in her local time.Wait, let me confirm with specific times.If the call starts at 5 PM A, it's 9 AM T, which is the start of the team's working hours.If it starts at 6 PM A, it's 10 AM T, which is still within their working hours.But if the call starts at 5:30 PM A, it's 9:30 AM T, which is fine.If it starts at 4:59 PM A, it's 8:59 AM T, which is before their working hours.Similarly, if it starts at 6:01 PM A, it's 10:01 AM T, which is still within their working hours until 6 PM T, which is 2 AM A.Wait, no. The team's working hours end at 6 PM T, which is 2 AM A. So, the call can go beyond 6 PM A as long as it's before 2 AM A. But the architect's working hours end at 6 PM A, so the call must end by 6 PM A.Wait, no, the call duration isn't specified, but the start time must be such that both are within their working hours.So, the call can start as late as 6 PM A, but it would end at 7 PM A, which is outside the architect's working hours. Therefore, the call must start by 6 PM A to end by 6 PM A, but that's only possible if the call duration is zero, which isn't practical.Wait, perhaps the call duration is negligible, or we're just considering the start time.But in any case, the latest start time is 6 PM A, but the team's working hours end at 6 PM T, which is 2 AM A. So, the call can start up to 6 PM A, but the team would still be working until 2 AM A.But the architect's working hours end at 6 PM A, so the call must start by 6 PM A to be within her working hours.But the team's working hours in A's time are 5 PM to 2 AM next day.So, the overlap between 5 PM to 2 AM and 9 AM to 6 PM is 5 PM to 6 PM.Therefore, the call must start between 5 PM and 6 PM A.So, the range is 5 PM to 6 PM.Therefore, the architect can schedule the call anytime between 5 PM and 6 PM in her local time.I think that's the correct answer.</think>

🧠:<think>Okay, so I have this problem about a coffee bean roaster who wants to create a blend using beans from three regions: A, B, and C. Each region's beans have a quality score that depends on the amount used, given by quadratic functions. The roaster has a budget constraint of 10 kg total. I need to figure out how to maximize the total quality score by choosing the right amounts from each region.First, let me break down the problem into two parts as given.Part 1: Derive the total quality score in terms of x and y.Alright, so the total quality score is the sum of the quality scores from each region. The functions are:- ( Q_A(x) = 3x^2 - 2x + 5 )- ( Q_B(y) = 4y^2 + y + 3 )- ( Q_C(z) = 2z^2 - 3z + 4 )And since the total beans used must add up to 10 kg, we know that ( x + y + z = 10 ). So, ( z = 10 - x - y ).So, to express the total quality score ( Q ) in terms of x and y, I need to substitute ( z ) in ( Q_C(z) ) with ( 10 - x - y ).Let me write that out:Total Quality Score ( Q = Q_A(x) + Q_B(y) + Q_C(z) )Substituting z:( Q = (3x^2 - 2x + 5) + (4y^2 + y + 3) + [2(10 - x - y)^2 - 3(10 - x - y) + 4] )Okay, so now I need to expand and simplify this expression.First, let's compute each part step by step.Compute ( Q_A(x) ):That's straightforward: ( 3x^2 - 2x + 5 )Compute ( Q_B(y) ):Also straightforward: ( 4y^2 + y + 3 )Compute ( Q_C(z) ):First, substitute z:( 2(10 - x - y)^2 - 3(10 - x - y) + 4 )Let me expand ( (10 - x - y)^2 ):( (10 - x - y)^2 = (10)^2 + (-x)^2 + (-y)^2 + 2*(10)*(-x) + 2*(10)*(-y) + 2*(-x)*(-y) )Wait, that might be too detailed. Alternatively, I can let ( a = 10 - x - y ), so ( a^2 = (10 - x - y)^2 ).But maybe it's easier to just expand it as:( (10 - x - y)^2 = (10)^2 + (-x - y)^2 + 2*(10)*(-x - y) )Wait, actually, no. The correct expansion is:( (a - b - c)^2 = a^2 + b^2 + c^2 - 2ab - 2ac + 2bc )Wait, actually, let me just compute it step by step.( (10 - x - y)^2 = (10)^2 + (-x)^2 + (-y)^2 + 2*(10)*(-x) + 2*(10)*(-y) + 2*(-x)*(-y) )So that's:( 100 + x^2 + y^2 - 20x - 20y + 2xy )So, ( (10 - x - y)^2 = x^2 + y^2 + 2xy - 20x - 20y + 100 )Therefore, ( Q_C(z) = 2*(x^2 + y^2 + 2xy - 20x - 20y + 100) - 3*(10 - x - y) + 4 )Let me compute each term:First term: ( 2*(x^2 + y^2 + 2xy - 20x - 20y + 100) )Multiply each term by 2:( 2x^2 + 2y^2 + 4xy - 40x - 40y + 200 )Second term: ( -3*(10 - x - y) )Multiply each term by -3:( -30 + 3x + 3y )Third term: +4So, combining all three terms:( 2x^2 + 2y^2 + 4xy - 40x - 40y + 200 - 30 + 3x + 3y + 4 )Simplify term by term:- Constants: 200 - 30 + 4 = 174- x terms: -40x + 3x = -37x- y terms: -40y + 3y = -37y- xy terms: 4xy- x² terms: 2x²- y² terms: 2y²So, ( Q_C(z) = 2x^2 + 2y^2 + 4xy - 37x - 37y + 174 )Now, let's go back to the total quality score:( Q = Q_A(x) + Q_B(y) + Q_C(z) )Substituting each part:( Q = (3x^2 - 2x + 5) + (4y^2 + y + 3) + (2x^2 + 2y^2 + 4xy - 37x - 37y + 174) )Now, let's combine like terms:First, let's list all the terms:From ( Q_A(x) ):- 3x²- -2x- +5From ( Q_B(y) ):- 4y²- +y- +3From ( Q_C(z) ):- 2x²- 2y²- 4xy- -37x- -37y- +174Now, combine term by term:x² terms: 3x² + 2x² = 5x²y² terms: 4y² + 2y² = 6y²xy terms: 4xyx terms: -2x -37x = -39xy terms: y -37y = -36yConstants: 5 + 3 + 174 = 182So, putting it all together:( Q = 5x² + 6y² + 4xy - 39x - 36y + 182 )So that's the total quality score in terms of x and y. That answers part 1.Part 2: Determine x and y that maximize Q, and find the maximum score.Alright, so now I have the total quality score as a function of x and y:( Q(x, y) = 5x² + 6y² + 4xy - 39x - 36y + 182 )I need to find the values of x and y that maximize this function, given that x, y, and z are non-negative (since you can't have negative beans) and x + y ≤ 10 (since z = 10 - x - y must be non-negative).So, this is a quadratic optimization problem with constraints. Since the function is quadratic and the constraints are linear, we can approach this by finding the critical point and then checking if it lies within the feasible region. If not, we check the boundaries.First, let's see if the function is concave or convex. Since the function is quadratic, the Hessian matrix will tell us. If the Hessian is negative definite, the function is concave, and the critical point is a maximum. If it's positive definite, it's convex, and the critical point is a minimum.Compute the Hessian matrix:The Hessian H is:[ d²Q/dx² d²Q/dxdy ][ d²Q/dydx d²Q/dy² ]Compute the second partial derivatives:d²Q/dx² = 10d²Q/dy² = 12d²Q/dxdy = d²Q/dydx = 4So, H = [10 4 4 12]To determine if H is positive definite, we can check the leading principal minors:First minor: 10 > 0Second minor: determinant of H = (10)(12) - (4)^2 = 120 - 16 = 104 > 0Since both leading principal minors are positive, H is positive definite. Therefore, the function is convex, and the critical point is a minimum.Wait, but we are supposed to maximize Q. If the function is convex, then the critical point is a minimum, which means the maximum must occur on the boundary of the feasible region.Hmm, that complicates things. So, the maximum won't be at the critical point but somewhere on the edges.But before jumping to conclusions, let me double-check my Hessian.Wait, the second derivatives:For Q(x, y) = 5x² + 6y² + 4xy - 39x - 36y + 182d²Q/dx² = 10d²Q/dy² = 12d²Q/dxdy = 4Yes, that's correct.So, H is positive definite, so the function is convex, meaning it has a unique minimum. Therefore, the maximum must occur on the boundary.So, to find the maximum, I need to check the boundaries of the feasible region.The feasible region is defined by:x ≥ 0y ≥ 0x + y ≤ 10So, the boundaries are:1. x = 02. y = 03. x + y = 10Additionally, within these boundaries, the maximum could be at the corners or along the edges.So, perhaps I should evaluate Q on each boundary and find the maximum.Alternatively, since the function is convex, the maximum will occur at one of the vertices of the feasible region.The feasible region is a triangle with vertices at (0,0), (10,0), and (0,10).So, perhaps I can evaluate Q at these three points and see which one gives the maximum.But wait, maybe the maximum is not necessarily at the vertices but somewhere along the edges. So, perhaps I need to check each edge as well.Alternatively, since the function is convex, the maximum will be at one of the vertices. Let me check that.Wait, actually, for convex functions over convex compact sets, the maximum is attained at an extreme point (vertex). So, in this case, since the feasible region is a convex polygon (a triangle), the maximum of Q will be at one of the vertices.Therefore, I can just evaluate Q at (0,0), (10,0), and (0,10), and see which is the maximum.But before I do that, let me compute Q at these points.First, (0,0):Q(0,0) = 5*(0)^2 + 6*(0)^2 + 4*(0)*(0) - 39*(0) - 36*(0) + 182 = 182Second, (10,0):Compute Q(10,0):= 5*(10)^2 + 6*(0)^2 + 4*(10)*(0) - 39*(10) - 36*(0) + 182= 5*100 + 0 + 0 - 390 - 0 + 182= 500 - 390 + 182= (500 - 390) + 182 = 110 + 182 = 292Third, (0,10):Compute Q(0,10):= 5*(0)^2 + 6*(10)^2 + 4*(0)*(10) - 39*(0) - 36*(10) + 182= 0 + 6*100 + 0 - 0 - 360 + 182= 600 - 360 + 182= 240 + 182 = 422So, among the vertices, Q(0,10) gives the highest value of 422.But wait, is that the maximum? Or could it be higher somewhere on the edges?Wait, since the function is convex, the maximum is indeed at the vertex. So, the maximum occurs at (0,10), giving Q=422.But let me double-check because sometimes intuition can be wrong.Alternatively, maybe I should check along each edge.Let me consider the edges:1. Edge where x=0, y varies from 0 to 10.On this edge, Q(0, y) = 5*0 + 6y² + 4*0*y - 39*0 - 36y + 182 = 6y² - 36y + 182This is a quadratic in y. Let's find its maximum.Since the coefficient of y² is positive, it's convex, so the minimum is at the vertex, and maximum occurs at the endpoints.So, endpoints are y=0 and y=10.At y=0: Q=182At y=10: Q=6*100 - 36*10 + 182=600 - 360 + 182=422So, maximum on this edge is 422 at y=10.2. Edge where y=0, x varies from 0 to 10.On this edge, Q(x,0)=5x² + 6*0 +4x*0 -39x -36*0 +182=5x² -39x +182Again, quadratic in x, coefficient of x² is positive, so convex. Maximum occurs at endpoints.At x=0: Q=182At x=10: Q=5*100 -39*10 +182=500 - 390 +182=292So, maximum on this edge is 292 at x=10.3. Edge where x + y =10, so y=10 -x, with x from 0 to10.On this edge, Q(x, 10 -x)=5x² +6*(10 -x)^2 +4x*(10 -x) -39x -36*(10 -x) +182Let me compute this step by step.First, expand each term:5x² remains as is.6*(10 -x)^2 =6*(100 -20x +x²)=600 -120x +6x²4x*(10 -x)=40x -4x²-39x remains as is.-36*(10 -x)= -360 +36x+182 remains as is.Now, combine all terms:5x² + (600 -120x +6x²) + (40x -4x²) -39x + (-360 +36x) +182Now, let's combine like terms:x² terms: 5x² +6x² -4x²=7x²x terms: -120x +40x -39x +36x= (-120 +40 -39 +36)x= (-120 +40= -80; -80 -39= -119; -119 +36= -83)xConstants: 600 -360 +182= (600 -360=240; 240 +182=422)So, Q(x,10 -x)=7x² -83x +422So, this is a quadratic in x, with coefficient of x² positive (7), so it's convex. Therefore, it has a minimum, and the maximum occurs at the endpoints.Endpoints are x=0 and x=10.At x=0: Q=7*0 -83*0 +422=422At x=10: Q=7*100 -83*10 +422=700 -830 +422= (700 -830)= -130 +422=292So, on this edge, maximum is 422 at x=0, which corresponds to y=10.So, all edges have their maximum at the vertices, and the overall maximum is at (0,10) with Q=422.Therefore, the maximum quality score is 422, achieved when x=0 kg from Region A, y=10 kg from Region B, and z=0 kg from Region C.Wait, but before finalizing, let me just think again. The function is convex, so its maximum over a convex set is at the vertex. So, the maximum is indeed at (0,10). Therefore, the roaster should use 10 kg from Region B and none from A and C to maximize the quality score.But just to be thorough, let me check if maybe somewhere inside the feasible region, even though it's a minimum, but perhaps the function could have higher values? Wait, no, since it's convex, the function curves upwards, so the minimum is in the middle, and the maximum is on the boundary.Therefore, yes, the maximum is at (0,10).Final AnswerThe maximum total quality score is achieved when using 0 kg from Region A, 10 kg from Region B, and 0 kg from Region C, resulting in a quality score of boxed{422}.</think>

🧠:<think>Okay, so I have this problem about RSA encryption, and I need to figure out the prime factors of n and then find the private key d. Let me start with Sub-problem 1.First, I know that in RSA, n is the product of two prime numbers, p and q. So, n = p * q. Here, n is given as 589. I need to find p and q such that when multiplied together, they give 589. Since 589 isn't a very large number, maybe I can factor it manually.Let me think about how to factor 589. I can start by checking divisibility by smaller prime numbers. Let's see:Is 589 even? No, because it ends with a 9, so it's not divisible by 2.Next, check divisibility by 3. The sum of the digits is 5 + 8 + 9 = 22. 22 isn't divisible by 3, so 589 isn't divisible by 3.How about 5? It doesn't end with a 0 or 5, so no.Next prime is 7. Let's divide 589 by 7. 7 times 80 is 560, subtracting that from 589 gives 29. 29 divided by 7 is about 4.14, which isn't an integer, so 7 doesn't divide 589.Next prime is 11. Let's try 11. 11 times 53 is 583, which is less than 589. 589 minus 583 is 6, so 11 doesn't divide 589.Next prime is 13. 13 times 45 is 585, which is 589 minus 585 is 4, so not divisible by 13.Next is 17. 17 times 34 is 578, subtracting that from 589 gives 11, so not divisible by 17.Next prime is 19. 19 times 31 is 589. Wait, let me check: 19*30 is 570, and 19 more is 589. Yes! So 19 times 31 is 589. Therefore, p and q are 19 and 31.Wait, let me confirm that 19*31 is indeed 589. 20*30 is 600, so 19*31 is 600 - 30 - 19 = 600 - 49 = 551? Wait, that doesn't make sense. Wait, maybe I miscalculated.Wait, 19*30 is 570, so 19*31 is 570 +19=589. Yes, that's correct. So p=19 and q=31.Okay, so Sub-problem 1 is solved. The prime factors are 19 and 31.Now, moving on to Sub-problem 2. I need to calculate the private key d such that e*d ≡ 1 mod φ(n). Given that e is 23, and n is 589, which we've factored into 19 and 31.First, I need to compute φ(n). Since n is the product of two distinct primes, φ(n) = (p-1)*(q-1). So, φ(589) = (19-1)*(31-1) = 18*30 = 540.So, φ(n) is 540. Now, I need to find d such that 23*d ≡ 1 mod 540. In other words, d is the multiplicative inverse of 23 modulo 540.To find d, I can use the Extended Euclidean Algorithm, which finds integers x and y such that 23x + 540y = 1. The x here will be the inverse of 23 modulo 540.Let me set up the algorithm.We need to find gcd(23, 540) and express it as a linear combination.First, divide 540 by 23:540 ÷ 23 = 23 with a remainder. Let me calculate 23*23=529. So, 540 - 529 = 11. So, 540 = 23*23 + 11.Now, take 23 and divide by the remainder 11:23 ÷ 11 = 2 with a remainder of 1, because 11*2=22, and 23-22=1.So, 23 = 11*2 + 1.Now, take 11 and divide by the remainder 1:11 ÷ 1 = 11 with remainder 0.So, the gcd is 1, which is expected since 23 is prime and doesn't divide 540.Now, working backwards to express 1 as a combination of 23 and 540.From the second step: 1 = 23 - 11*2.But 11 is from the first step: 11 = 540 - 23*23.Substitute that into the equation:1 = 23 - (540 - 23*23)*2.Let me expand that:1 = 23 - 540*2 + 23*46.Combine like terms:1 = 23*(1 + 46) - 540*2.1 = 23*47 - 540*2.So, this gives us 1 = 23*47 - 540*2.Therefore, x is 47 and y is -2.So, the inverse of 23 modulo 540 is 47. But let me check that.23*47 = 1081. Now, divide 1081 by 540: 540*2=1080, so 1081-1080=1. So, yes, 23*47 ≡ 1 mod 540.Therefore, d is 47.Wait, but let me confirm once more. 23*47 is 1081. 1081 divided by 540 is 2 with a remainder of 1. So, yes, 23*47 ≡ 1 mod 540. So, d=47.So, the private key is d=47.Wait, but let me make sure I didn't make a mistake in the Extended Euclidean steps.Starting with 540 and 23.540 = 23*23 + 1123 = 11*2 + 111 = 1*11 + 0So, gcd is 1.Then, backtracking:1 = 23 - 11*2But 11 = 540 - 23*23So, 1 = 23 - (540 - 23*23)*2= 23 - 540*2 + 23*46= 23*(1 + 46) - 540*2= 23*47 - 540*2Yes, that's correct. So, x=47, which is the inverse.So, d=47.I think that's correct.Final AnswerSub-problem 1: The prime factors are boxed{19} and boxed{31}.Sub-problem 2: The private key is boxed{47}.</think>