🧠:<think>Okay, so I have this problem about the Huskies football team and their games over the years. Let me try to break it down step by step. First, the problem says that the Huskies started playing in 1922, and they've played every year without interruption up to 2023. The number of games they played each year forms an arithmetic sequence. In 1922, they played 5 games, and in 2023, they played 12 games. I need to find two things: the common difference (d) of this arithmetic sequence and the total number of games played from 1922 to 2023. Alright, let's start with the first part: finding the common difference (d). An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference. The formula for the (n)-th term of an arithmetic sequence is:[a_n = a_1 + (n - 1)d]Here, (a_n) is the (n)-th term, (a_1) is the first term, (d) is the common difference, and (n) is the term number.In this case, the first term (a_1) is 5 games in 1922. The last term given is 12 games in 2023. So, I need to figure out how many years are between 1922 and 2023, inclusive, to determine the value of (n).Let me calculate the number of years. From 1922 to 2023, that's 2023 - 1922 + 1 years because both 1922 and 2023 are included. Calculating that: 2023 - 1922 = 101, so adding 1 gives 102 years. So, (n = 102).Now, plugging into the formula for the (n)-th term:[a_{102} = a_1 + (102 - 1)d]We know that (a_{102} = 12) and (a_1 = 5). So,[12 = 5 + 101d]Subtracting 5 from both sides:[7 = 101d]So, solving for (d):[d = frac{7}{101}]Hmm, that gives me a fractional common difference. Is that okay? Well, in real life, the number of games should be whole numbers each year, but since the problem states it's an arithmetic sequence, I guess it's acceptable for the difference to be a fraction. So, (d = frac{7}{101}). Wait, let me double-check my calculation. The number of terms is 102 because from 1922 to 2023 inclusive is 102 years. So, 2023 - 1922 = 101, plus 1 is 102. That seems right. So, the common difference is 7 divided by 101, which is approximately 0.0693. So, each year, the number of games increases by about 0.0693. That seems like a very small increase, but over 101 years, it adds up to 7 games. Okay, moving on to the second part: calculating the total number of games played from 1922 to 2023. The formula for the sum of the first (n) terms of an arithmetic sequence is:[S_n = frac{n}{2} (a_1 + a_n)]Where (S_n) is the sum, (n) is the number of terms, (a_1) is the first term, and (a_n) is the last term.We have (n = 102), (a_1 = 5), and (a_n = 12). Plugging these into the formula:[S_{102} = frac{102}{2} (5 + 12)]Simplifying:[S_{102} = 51 times 17]Calculating 51 multiplied by 17. Let me do that step by step:50 times 17 is 850, and 1 times 17 is 17, so 850 + 17 = 867.So, the total number of games is 867.Wait, that seems straightforward. Let me confirm if I used the correct formula. Yes, the sum of an arithmetic series is indeed the average of the first and last term multiplied by the number of terms. So, 5 + 12 is 17, divided by 2 is 8.5, multiplied by 102 gives 867. Yep, that's correct.But just to make sure, let me also try using the other formula for the sum:[S_n = frac{n}{2} [2a_1 + (n - 1)d]]We already know (d = frac{7}{101}), so plugging in:[S_{102} = frac{102}{2} [2 times 5 + (102 - 1) times frac{7}{101}]]Simplify:[S_{102} = 51 [10 + 101 times frac{7}{101}]]Simplify inside the brackets:101 divided by 101 is 1, so 101 times 7/101 is 7.So, inside the brackets, it's 10 + 7 = 17.Therefore,[S_{102} = 51 times 17 = 867]Same result. So, that confirms it. The total number of games is 867.So, summarizing:1. The common difference (d) is (frac{7}{101}).2. The total number of games is 867.Just to recap, the steps were:1. Determine the number of terms, which is the number of years from 1922 to 2023 inclusive, which is 102.2. Use the arithmetic sequence formula to find the common difference (d).3. Use the sum formula to find the total number of games.Everything seems to check out. I don't see any mistakes in my calculations, so I think that's the correct answer.Final Answer1. The common difference (d) is boxed{dfrac{7}{101}}.2. The total number of games played is boxed{867}.</think>

🧠:<think>Okay, so I have this problem about an entrepreneur running a health-conscious restaurant. They're worried about a private chef who's becoming popular, and they want to figure out how to stay competitive. The problem has two parts: one about ingredient costs and another about customer satisfaction. Let me tackle them one by one.Starting with the first part: Ingredient Costs. The restaurant uses five different organic ingredients for their signature dish. Each ingredient costs x_i per unit, and they use a_i units per dish. The dish sells for P dollars. The private chef's costs are 10% lower on average, and the restaurant is concerned about maintaining a profit margin of at least 20%. Also, there's a potential 15% increase in ingredient costs due to market fluctuations. I need to derive an expression for the minimum price P_{text{min}} the restaurant must set.Alright, so first, let's figure out the current cost per dish. The cost for each ingredient is x_i per unit, and they use a_i units. So, the total cost for ingredient i is x_i times a_i. Since there are five ingredients, the total cost for all ingredients per dish is the sum from i=1 to 5 of x_i a_i. Let me write that as:text{Total Cost} = sum_{i=1}^{5} x_i a_iNow, the private chef's costs are 10% lower. That means the private chef's cost per unit is 0.9x_i for each ingredient. So, their total cost per dish would be:text{Private Chef's Total Cost} = sum_{i=1}^{5} 0.9x_i a_iBut wait, the restaurant is worried about their own costs increasing by 15% due to market fluctuations. So, the restaurant's cost could go up by 15%. Therefore, the new cost per dish for the restaurant would be:text{New Total Cost} = sum_{i=1}^{5} x_i a_i times 1.15So, the restaurant's cost becomes 1.15 times their original cost.The restaurant wants to maintain a profit margin of at least 20%. Profit margin is calculated as (Profit / Revenue) * 100%. So, if they want a 20% profit margin, their profit should be 20% of their revenue.Let me denote the selling price as P. Then, the revenue per dish is P, and the cost is 1.15 times the original total cost. So, the profit per dish is:text{Profit} = P - text{New Total Cost}We want this profit to be at least 20% of the revenue, so:P - text{New Total Cost} geq 0.20 PSimplifying this inequality:P - 0.20 P geq text{New Total Cost}0.80 P geq text{New Total Cost}Therefore, solving for P:P geq frac{text{New Total Cost}}{0.80}Substituting the expression for New Total Cost:P geq frac{1.15 times sum_{i=1}^{5} x_i a_i}{0.80}So, the minimum price P_{text{min}} is:P_{text{min}} = frac{1.15}{0.80} times sum_{i=1}^{5} x_i a_iSimplifying the constants:1.15 divided by 0.80 is equal to 1.4375. So,P_{text{min}} = 1.4375 times sum_{i=1}^{5} x_i a_iWait, let me check that division again. 1.15 divided by 0.8. Hmm, 0.8 goes into 1.15 once, with 0.35 remaining. 0.35 divided by 0.8 is 0.4375. So, yes, 1.4375 is correct.So, that's the expression for the minimum price. It's 1.4375 times the sum of each ingredient's cost times the amount used per dish.Moving on to the second part: Customer Satisfaction. The entrepreneur models customer satisfaction S(t) as a function of time t (in months) since the restaurant opened. The differential equation given is:frac{dS}{dt} = k(S_m - S) - frac{C}{1 + e^{-rt}}We need to solve this differential equation with the initial condition S(0) = S_0, and then discuss how the entrepreneur can adjust k or C to counteract the increasing popularity of the private chef.Alright, so this is a first-order linear ordinary differential equation. The standard form is:frac{dS}{dt} + P(t) S = Q(t)Let me rewrite the given equation to match this form.Starting with:frac{dS}{dt} = k(S_m - S) - frac{C}{1 + e^{-rt}}Let me distribute the k:frac{dS}{dt} = k S_m - k S - frac{C}{1 + e^{-rt}}Now, bring all terms involving S to the left:frac{dS}{dt} + k S = k S_m - frac{C}{1 + e^{-rt}}So, in standard form, P(t) = k and Q(t) = k S_m - frac{C}{1 + e^{-rt}}.To solve this, we can use an integrating factor. The integrating factor mu(t) is:mu(t) = e^{int P(t) dt} = e^{int k dt} = e^{k t}Multiplying both sides of the differential equation by mu(t):e^{k t} frac{dS}{dt} + k e^{k t} S = e^{k t} left( k S_m - frac{C}{1 + e^{-rt}} right )The left side is the derivative of S(t) e^{k t}:frac{d}{dt} left( S e^{k t} right ) = e^{k t} left( k S_m - frac{C}{1 + e^{-rt}} right )Now, integrate both sides with respect to t:S e^{k t} = int e^{k t} left( k S_m - frac{C}{1 + e^{-rt}} right ) dt + DWhere D is the constant of integration.Let me split the integral into two parts:S e^{k t} = k S_m int e^{k t} dt - C int frac{e^{k t}}{1 + e^{-rt}} dt + DCompute the first integral:int e^{k t} dt = frac{e^{k t}}{k} + C_1So, the first term becomes:k S_m times frac{e^{k t}}{k} = S_m e^{k t}Now, the second integral is more complicated:int frac{e^{k t}}{1 + e^{-rt}} dtLet me simplify the denominator:1 + e^{-rt} = frac{e^{rt} + 1}{e^{rt}}So, the integral becomes:int frac{e^{k t} times e^{rt}}{e^{rt} + 1} dt = int frac{e^{(k + r) t}}{e^{rt} + 1} dtLet me make a substitution. Let u = e^{rt} + 1, then du/dt = r e^{rt}, so du = r e^{rt} dt. Hmm, but we have e^{(k + r)t} in the numerator.Let me write e^{(k + r)t} = e^{rt} times e^{kt}. So, the integral becomes:int frac{e^{rt} e^{kt}}{e^{rt} + 1} dt = int frac{e^{kt}}{1 + e^{-rt}} e^{rt} dtWait, maybe another substitution. Let me set u = e^{rt}, so du = r e^{rt} dt, which is du = r u dt, so dt = du/(r u).Then, the integral becomes:int frac{e^{kt}}{1 + e^{-rt}} dt = int frac{e^{kt}}{1 + 1/u} times frac{du}{r u}Simplify the denominator:1 + 1/u = (u + 1)/u, so:int frac{e^{kt}}{(u + 1)/u} times frac{du}{r u} = int frac{e^{kt} u}{u + 1} times frac{du}{r u} = frac{1}{r} int frac{e^{kt}}{u + 1} duBut u = e^{rt}, so e^{kt} = e^{k t} = e^{k times ( ln u / r ) } = u^{k/r}.So, substituting back:frac{1}{r} int frac{u^{k/r}}{u + 1} duThis integral is still complicated. Maybe another approach.Alternatively, let me consider the substitution v = e^{rt}, so dv = r e^{rt} dt, so dt = dv/(r e^{rt}) = dv/(r v).Then, the integral becomes:int frac{e^{kt}}{1 + e^{-rt}} dt = int frac{e^{kt}}{1 + 1/v} times frac{dv}{r v}Simplify the denominator:1 + 1/v = (v + 1)/v, so:int frac{e^{kt} v}{v + 1} times frac{dv}{r v} = frac{1}{r} int frac{e^{kt}}{v + 1} dvBut v = e^{rt}, so e^{kt} = e^{k t} = e^{k times (ln v / r)} = v^{k/r}.Thus, the integral becomes:frac{1}{r} int frac{v^{k/r}}{v + 1} dvThis integral is still not straightforward. Maybe another substitution? Let me set w = v + 1, so dw = dv, and v = w - 1. Then,frac{1}{r} int frac{(w - 1)^{k/r}}{w} dwThis might not help much either. Perhaps there's a better way. Maybe express the integrand as a series expansion?Alternatively, perhaps recognize that the integral can be expressed in terms of the exponential integral function, but that might be beyond the scope here.Wait, maybe I made a mistake earlier. Let me try a different substitution.Let me set z = e^{(k + r)t}, so dz/dt = (k + r) e^{(k + r)t}, so dt = dz / [(k + r) z].Then, the integral becomes:int frac{e^{(k + r)t}}{e^{rt} + 1} dt = int frac{z}{e^{rt} + 1} times frac{dz}{(k + r) z}But e^{rt} = e^{rt} = z^{r/(k + r)} since z = e^{(k + r)t} implies e^{rt} = z^{r/(k + r)}.So, the integral becomes:frac{1}{k + r} int frac{1}{z^{r/(k + r)} + 1} dzThis is still complicated. Maybe it's better to look for another approach.Alternatively, perhaps consider that the term frac{C}{1 + e^{-rt}} is a logistic function, which asymptotically approaches C as t increases. So, maybe the integral can be expressed in terms of logarithmic functions or something similar.Wait, let me consider integrating frac{e^{kt}}{1 + e^{-rt}} dt.Let me make substitution u = e^{-rt}, so du/dt = -r e^{-rt}, so dt = -du/(r u).Then, the integral becomes:int frac{e^{kt}}{1 + u} times left( -frac{du}{r u} right ) = -frac{1}{r} int frac{e^{kt}}{u(1 + u)} duBut e^{kt} = e^{kt} = e^{k t} = e^{k times (-ln u / r)} = u^{-k/r}.So, substituting:-frac{1}{r} int frac{u^{-k/r}}{u(1 + u)} du = -frac{1}{r} int frac{u^{-(k/r + 1)}}{1 + u} duLet me set n = -(k/r + 1), so the integral becomes:-frac{1}{r} int u^{n} / (1 + u) duThis integral can be expressed in terms of the digamma function or hypergeometric functions, but perhaps it's better to leave it in terms of logarithms or use partial fractions.Alternatively, perhaps consider expanding 1/(1 + u) as a series if u is small, but that might not be valid for all t.Hmm, this seems getting too complicated. Maybe I need to look for another method or perhaps recognize that this integral doesn't have an elementary closed-form solution.Wait, maybe I can express the integral as:int frac{e^{kt}}{1 + e^{-rt}} dt = int frac{e^{(k + r)t}}{e^{rt} + 1} dtLet me set u = e^{rt}, so du = r e^{rt} dt, so dt = du/(r u).Then, the integral becomes:int frac{e^{(k + r)t}}{u + 1} times frac{du}{r u}But e^{(k + r)t} = e^{(k + r) times (ln u / r)} = u^{(k + r)/r} = u^{1 + k/r}.So, substituting:frac{1}{r} int frac{u^{1 + k/r}}{u(u + 1)} du = frac{1}{r} int frac{u^{k/r}}{u + 1} duThis is similar to what I had earlier. Maybe express this as:frac{1}{r} int frac{u^{k/r}}{u + 1} duThis integral is known and can be expressed in terms of the digamma function or the logarithmic integral, but perhaps for the purposes of this problem, we can leave it as is or express it in terms of a special function.Alternatively, perhaps we can write it as:frac{1}{r} left( frac{u^{k/r + 1}}{k/r + 1} - int frac{u^{k/r + 1}}{(u + 1)(k/r + 1)} du right )But that doesn't seem helpful.Wait, maybe another substitution. Let me set w = u + 1, so dw = du, and u = w - 1. Then,frac{1}{r} int frac{(w - 1)^{k/r}}{w} dwThis is still complicated, but perhaps we can expand (w - 1)^{k/r} using the binomial theorem if k/r is an integer, but it's not specified.Given that this integral is complicated, perhaps it's better to express the solution in terms of an integral that cannot be simplified further. So, going back to the expression:S e^{k t} = S_m e^{k t} - C int frac{e^{(k + r)t}}{e^{rt} + 1} dt + DWait, actually, earlier I had:S e^{k t} = S_m e^{k t} - C int frac{e^{(k + r)t}}{e^{rt} + 1} dt + DSo, to solve for S(t), we can write:S(t) = S_m - C e^{-k t} int frac{e^{(k + r)t}}{e^{rt} + 1} dt + D e^{-k t}Now, applying the initial condition S(0) = S_0:At t = 0,S(0) = S_0 = S_m - C e^{0} int_{0}^{0} frac{e^{(k + r) cdot 0}}{e^{0} + 1} dt + D e^{0}Wait, actually, the integral from 0 to 0 is zero, so:S_0 = S_m - C times 0 + DThus, D = S_0 - S_m.So, substituting back:S(t) = S_m - C e^{-k t} int_{0}^{t} frac{e^{(k + r)s}}{e^{r s} + 1} ds + (S_0 - S_m) e^{-k t}This is the general solution. It might not have a simpler closed-form expression unless we can evaluate the integral in terms of elementary functions, which seems challenging.Alternatively, perhaps we can express the integral in terms of logarithms or other functions. Let me try another substitution for the integral:Let u = e^{r s}, so du = r e^{r s} ds, so ds = du/(r u).Then, the integral becomes:int frac{e^{(k + r)s}}{e^{r s} + 1} ds = int frac{e^{(k + r)s}}{u + 1} times frac{du}{r u}But e^{(k + r)s} = e^{(k + r)s} = u^{(k + r)/r} = u^{1 + k/r}.So, substituting:int frac{u^{1 + k/r}}{u(u + 1)} times frac{du}{r} = frac{1}{r} int frac{u^{k/r}}{u + 1} duThis is the same integral as before. So, perhaps we can express it as:frac{1}{r} left( frac{u^{k/r + 1}}{k/r + 1} - int frac{u^{k/r + 1}}{(u + 1)(k/r + 1)} du right )But this doesn't seem helpful. Alternatively, perhaps express it as:frac{1}{r} int frac{u^{k/r}}{u + 1} du = frac{1}{r} left( int frac{u^{k/r}}{u + 1} du right )This integral can be expressed in terms of the digamma function or the logarithmic integral, but for the purposes of this problem, maybe we can leave it as is.Alternatively, perhaps recognize that for certain values of k and r, this integral can be simplified. For example, if k = r, then the integral becomes:frac{1}{r} int frac{u^{1}}{u + 1} du = frac{1}{r} int left( 1 - frac{1}{u + 1} right ) du = frac{1}{r} left( u - ln(u + 1) right ) + CBut since k and r are constants, unless specified, we can't assume they are equal. So, perhaps the best we can do is express the solution in terms of this integral.Therefore, the solution is:S(t) = S_m - C e^{-k t} int_{0}^{t} frac{e^{(k + r)s}}{e^{r s} + 1} ds + (S_0 - S_m) e^{-k t}Alternatively, we can write it as:S(t) = S_m + (S_0 - S_m) e^{-k t} - C e^{-k t} int_{0}^{t} frac{e^{(k + r)s}}{e^{r s} + 1} dsThis is the expression for S(t). Now, the entrepreneur wants to discuss how to adjust k or C to counteract the increasing popularity of the private chef.The private chef's popularity might be leading to a decrease in customer satisfaction at the restaurant, or perhaps an increase in competition, which could affect the restaurant's satisfaction scores. The differential equation models the change in satisfaction over time.The term k(S_m - S) suggests that the satisfaction tends to approach S_m over time, with k controlling the rate of approach. A higher k would mean faster convergence to S_m. The term - frac{C}{1 + e^{-rt}} represents some external influence on satisfaction, perhaps initial interest or marketing efforts that decay over time as t increases.If the private chef is becoming more popular, perhaps the restaurant's satisfaction is decreasing, or the rate of approach to S_m is slowing down. To counteract this, the entrepreneur might need to increase k to make the satisfaction approach S_m faster, thereby maintaining higher satisfaction levels. Alternatively, increasing C could amplify the external influence term, perhaps through better marketing or improved customer engagement, which could help offset the negative impact of the private chef's rise in popularity.Alternatively, if the term - frac{C}{1 + e^{-rt}} is causing a decrease in satisfaction, perhaps the entrepreneur could adjust C to be negative, effectively adding a positive term that boosts satisfaction. However, since C is given as a positive constant representing initial customer interest, making it negative might not be appropriate. Instead, increasing C would mean a stronger initial influence, which might help in the short term, but as t increases, the effect diminishes.Alternatively, adjusting k to be larger would mean that the restaurant's satisfaction recovers more quickly to S_m, which could help if the private chef is causing a temporary dip in satisfaction.In summary, to counteract the increasing popularity of the private chef, the entrepreneur could either increase k to accelerate the return to maximum satisfaction or increase C to enhance the external positive influence on customer satisfaction, though the latter effect diminishes over time.So, putting it all together, the minimum price P_{text{min}} is 1.4375 times the total ingredient cost, and the customer satisfaction model suggests adjusting k or C to maintain satisfaction levels despite competition.</think>

🧠:<think>Alright, so I have this problem about orbital mechanics, and I need to figure it out step by step. Let me start by reading the problem again to make sure I understand what's being asked.First, there's a space enthusiast analyzing an exoplanet's orbit. The exoplanet orbits a star with mass ( M ), and the orbit is elliptical with semi-major axis ( a ) and eccentricity ( e ). The first part asks me to derive the equation for the orbital period ( T ) using Kepler's third law, expressed in terms of ( a ) and ( M ). Okay, I remember Kepler's third law relates the orbital period to the semi-major axis and the mass of the star. I think it's something like ( T^2 ) is proportional to ( a^3 ), but I need to recall the exact formula.Then, the second part gives specific values: ( a = 1.5 times 10^{11} ) meters, ( e = 0.1 ), and ( M = 2 times 10^{30} ) kg. I need to calculate the velocity at perihelion and aphelion. Hmm, velocities at the closest and farthest points in the orbit. I think that involves the vis-viva equation, which relates the velocity at any point in the orbit to the semi-major axis, eccentricity, and the distance from the star at that point.Let me tackle the first part first.Part 1: Deriving Kepler's Third LawKepler's third law states that the square of the orbital period ( T ) is proportional to the cube of the semi-major axis ( a ) of the orbit. Mathematically, this is:[ T^2 propto a^3 ]But to express it in terms of ( M ), the mass of the star, I need to include Newton's version of Kepler's third law, which incorporates gravitational constants. Newton generalized Kepler's law to include the masses involved.The formula is:[ T^2 = frac{4pi^2}{G M} a^3 ]Where:- ( G ) is the gravitational constant (( 6.674 times 10^{-11} , text{m}^3 text{kg}^{-1} text{s}^{-2} ))- ( M ) is the mass of the star- ( a ) is the semi-major axisSo, solving for ( T ), the orbital period is:[ T = 2pi sqrt{frac{a^3}{G M}} ]That should be the equation for the orbital period. Let me just verify that the units make sense. ( a ) is in meters, ( M ) in kilograms, ( G ) has units of m³ kg⁻¹ s⁻². So ( a^3 ) is m³, divided by ( G M ) gives (m³)/(m³ kg⁻¹ s⁻²) = s². Taking the square root gives seconds, which is correct for time. Multiplying by ( 2pi ) keeps it in seconds. Yep, that seems right.Part 2: Calculating Velocities at Perihelion and AphelionAlright, now for the velocities. I remember that in an elliptical orbit, the velocity varies with the distance from the star. The velocities at perihelion (closest point) and aphelion (farthest point) can be found using the vis-viva equation.The vis-viva equation is:[ v = sqrt{G M left( frac{2}{r} - frac{1}{a} right)} ]Where:- ( r ) is the distance from the star at the point of interest- ( a ) is the semi-major axis- ( G ) and ( M ) are the same as beforeAt perihelion, ( r = a (1 - e) ), and at aphelion, ( r = a (1 + e) ).Given ( a = 1.5 times 10^{11} ) m and ( e = 0.1 ), let's compute these distances first.Perihelion distance ( r_p = a (1 - e) = 1.5 times 10^{11} times (1 - 0.1) = 1.5 times 10^{11} times 0.9 = 1.35 times 10^{11} ) meters.Aphelion distance ( r_a = a (1 + e) = 1.5 times 10^{11} times 1.1 = 1.65 times 10^{11} ) meters.Now, plug these into the vis-viva equation.First, let's compute the constants:( G = 6.674 times 10^{-11} , text{m}^3 text{kg}^{-1} text{s}^{-2} )( M = 2 times 10^{30} ) kgCompute ( G M ):( G M = 6.674 times 10^{-11} times 2 times 10^{30} = 13.348 times 10^{19} ) m³ s⁻²Wait, let me compute that step by step:( 6.674 times 2 = 13.348 )So, ( 13.348 times 10^{-11 + 30} = 13.348 times 10^{19} ) m³ s⁻².Okay, so ( G M = 1.3348 times 10^{20} ) m³ s⁻².Wait, actually, 13.348 x 10^19 is 1.3348 x 10^20. Yes, that's correct.Now, let's compute the terms inside the square root for perihelion and aphelion.Starting with perihelion:( v_p = sqrt{G M left( frac{2}{r_p} - frac{1}{a} right)} )Compute ( frac{2}{r_p} ):( frac{2}{1.35 times 10^{11}} = frac{2}{1.35} times 10^{-11} approx 1.4815 times 10^{-11} ) m⁻¹Compute ( frac{1}{a} ):( frac{1}{1.5 times 10^{11}} = frac{1}{1.5} times 10^{-11} approx 0.6667 times 10^{-11} ) m⁻¹Subtract the two:( 1.4815 times 10^{-11} - 0.6667 times 10^{-11} = (1.4815 - 0.6667) times 10^{-11} approx 0.8148 times 10^{-11} ) m⁻¹Multiply by ( G M ):( 1.3348 times 10^{20} times 0.8148 times 10^{-11} = 1.3348 times 0.8148 times 10^{9} )Calculate 1.3348 * 0.8148:Let me compute 1.3348 * 0.8 = 1.067841.3348 * 0.0148 ≈ 0.0197So total ≈ 1.06784 + 0.0197 ≈ 1.0875So, approximately 1.0875 x 10^9 m² s⁻²Take the square root:( sqrt{1.0875 times 10^9} approx sqrt{1.0875} times 10^{4.5} )Compute sqrt(1.0875): approximately 1.042810^4.5 is sqrt(10^9) = 31622.7766So, 1.0428 * 31622.7766 ≈ 32940 m/sWait, that seems high. Let me check my calculations again.Wait, maybe I messed up the exponents.Wait, 1.3348 x 10^20 * 0.8148 x 10^-11 = (1.3348 * 0.8148) x 10^(20 - 11) = 1.0875 x 10^9 m²/s².Yes, that's correct. Then sqrt(1.0875 x 10^9) = sqrt(1.0875) x 10^4.5.Wait, 10^4.5 is 10^(4 + 0.5) = 10^4 * 10^0.5 ≈ 10000 * 3.1623 ≈ 31623.So sqrt(1.0875) ≈ 1.0428, so 1.0428 * 31623 ≈ 32940 m/s.Wait, that seems a bit high for orbital velocity, but let's see.Alternatively, maybe I should compute it more accurately.Let me compute 1.3348 * 0.8148:1.3348 * 0.8 = 1.067841.3348 * 0.0148 = let's compute 1.3348 * 0.01 = 0.0133481.3348 * 0.0048 = approx 0.006407So total 0.013348 + 0.006407 ≈ 0.019755So total 1.06784 + 0.019755 ≈ 1.087595So 1.087595 x 10^9 m²/s².So sqrt(1.087595 x 10^9) = sqrt(1.087595) x sqrt(10^9) ≈ 1.0428 x 31622.7766 ≈ 32940 m/s.Hmm, that seems correct, but let me cross-verify.Alternatively, maybe I can compute it as:sqrt(1.087595e9) = sqrt(1.087595) * sqrt(1e9) ≈ 1.0428 * 31622.7766 ≈ 32940 m/s.Alternatively, perhaps I should compute it more precisely.Wait, 1.087595e9 is 1,087,595,000.Compute sqrt(1,087,595,000). Let's see, sqrt(1,000,000,000) is 31,622.7766.So, 1,087,595,000 is 1.087595 times 1e9, so sqrt is sqrt(1.087595) * 31,622.7766.Compute sqrt(1.087595):Let me compute 1.0428^2 = approx 1.0875, which matches. So yes, 1.0428 * 31,622.7766 ≈ 32,940 m/s.Okay, so perihelion velocity is approximately 32,940 m/s.Now, let's compute the aphelion velocity.Using the same vis-viva equation:( v_a = sqrt{G M left( frac{2}{r_a} - frac{1}{a} right)} )Compute ( frac{2}{r_a} ):( frac{2}{1.65 times 10^{11}} = frac{2}{1.65} times 10^{-11} ≈ 1.2121 times 10^{-11} ) m⁻¹Compute ( frac{1}{a} ):Same as before, ( 0.6667 times 10^{-11} ) m⁻¹Subtract:( 1.2121 times 10^{-11} - 0.6667 times 10^{-11} = (1.2121 - 0.6667) times 10^{-11} ≈ 0.5454 times 10^{-11} ) m⁻¹Multiply by ( G M ):( 1.3348 times 10^{20} times 0.5454 times 10^{-11} = 1.3348 times 0.5454 times 10^{9} )Compute 1.3348 * 0.5454:1.3348 * 0.5 = 0.66741.3348 * 0.0454 ≈ 0.0607So total ≈ 0.6674 + 0.0607 ≈ 0.7281So, approximately 0.7281 x 10^9 m²/s²Take the square root:sqrt(0.7281 x 10^9) = sqrt(0.7281) x 10^4.5sqrt(0.7281) ≈ 0.853310^4.5 ≈ 31622.7766So, 0.8533 * 31622.7766 ≈ 27,000 m/sWait, let me compute it more accurately.Compute 0.7281 x 10^9 = 728,100,000.sqrt(728,100,000) = sqrt(728.1 x 10^6) = sqrt(728.1) x 10^3.sqrt(728.1) ≈ 26.98, because 27^2 = 729, so sqrt(728.1) ≈ 26.98.So, 26.98 x 10^3 ≈ 26,980 m/s.So, approximately 26,980 m/s.Wait, that seems a bit low, but considering it's the aphelion, which is farther away, the velocity should be lower than perihelion, which was about 32,940 m/s. So that makes sense.Alternatively, let me compute it step by step.Compute 1.3348 * 0.5454:1.3348 * 0.5 = 0.66741.3348 * 0.04 = 0.0533921.3348 * 0.0054 ≈ 0.007218Adding up: 0.6674 + 0.053392 + 0.007218 ≈ 0.72801So, 0.72801 x 10^9 m²/s².sqrt(0.72801e9) = sqrt(0.72801) x 10^4.5 ≈ 0.8533 x 31622.7766 ≈ 26,980 m/s.Yes, that's correct.So, summarizing:Perihelion velocity ( v_p ≈ 32,940 ) m/sAphelion velocity ( v_a ≈ 26,980 ) m/sWait, but let me check if I did the calculations correctly, because sometimes when dealing with exponents, it's easy to make a mistake.Wait, another approach: compute ( G M ) as 1.3348e20 m³/s².Then, for perihelion:( v_p = sqrt{1.3348e20 times (2/(1.35e11) - 1/(1.5e11))} )Compute inside the brackets:2/(1.35e11) = approx 1.4815e-111/(1.5e11) = approx 6.6667e-12Subtract: 1.4815e-11 - 6.6667e-12 = (1.4815 - 0.6667)e-11 = 0.8148e-11 = 8.148e-12Multiply by G M: 1.3348e20 * 8.148e-12 = (1.3348 * 8.148)e8Compute 1.3348 * 8.148:1 * 8.148 = 8.1480.3348 * 8.148 ≈ 2.726Total ≈ 8.148 + 2.726 ≈ 10.874So, 10.874e8 = 1.0874e9 m²/s²sqrt(1.0874e9) ≈ 32,970 m/s, which is close to my earlier result.Similarly, for aphelion:Inside the brackets: 2/(1.65e11) = approx 1.2121e-111/(1.5e11) = 6.6667e-12Subtract: 1.2121e-11 - 6.6667e-12 = (1.2121 - 0.6667)e-11 = 0.5454e-11 = 5.454e-12Multiply by G M: 1.3348e20 * 5.454e-12 = (1.3348 * 5.454)e8Compute 1.3348 * 5.454:1 * 5.454 = 5.4540.3348 * 5.454 ≈ 1.826Total ≈ 5.454 + 1.826 ≈ 7.280So, 7.280e8 = 7.28e8 m²/s²sqrt(7.28e8) ≈ 26,980 m/s, which matches my earlier calculation.So, I think these velocities are correct.Wait, but let me just cross-verify using another method. I remember that the product of the velocities at perihelion and aphelion equals the square of the circular velocity at the semi-major axis. Wait, is that correct?Wait, no, actually, in an elliptical orbit, the specific orbital energy is given by ( epsilon = - frac{G M}{2 a} ). The specific angular momentum ( h ) is ( sqrt{G M a (1 - e^2)} ).The velocities at perihelion and aphelion can also be expressed as:( v_p = sqrt{frac{G M}{a} left( frac{1 + e}{1 - e} right)} )( v_a = sqrt{frac{G M}{a} left( frac{1 - e}{1 + e} right)} )Wait, let me check that formula.Yes, because at perihelion, ( r = a (1 - e) ), and using the vis-viva equation:( v_p = sqrt{G M left( frac{2}{a (1 - e)} - frac{1}{a} right)} = sqrt{G M left( frac{2 - (1 - e)}{a (1 - e)} right)} = sqrt{G M left( frac{1 + e}{a (1 - e)} right)} = sqrt{frac{G M (1 + e)}{a (1 - e)}} )Similarly, for aphelion:( v_a = sqrt{frac{G M (1 - e)}{a (1 + e)}} )So, let's compute these.Given ( G M = 1.3348e20 ) m³/s², ( a = 1.5e11 ) m, ( e = 0.1 ).Compute ( v_p = sqrt{frac{1.3348e20 times (1 + 0.1)}{1.5e11 times (1 - 0.1)}} )Simplify numerator and denominator:Numerator: 1.3348e20 * 1.1 = 1.46828e20Denominator: 1.5e11 * 0.9 = 1.35e11So, ( v_p = sqrt{frac{1.46828e20}{1.35e11}} = sqrt{1.0876e9} ≈ 32,980 ) m/sSimilarly, ( v_a = sqrt{frac{1.3348e20 times (1 - 0.1)}{1.5e11 times (1 + 0.1)}} )Numerator: 1.3348e20 * 0.9 = 1.20132e20Denominator: 1.5e11 * 1.1 = 1.65e11So, ( v_a = sqrt{frac{1.20132e20}{1.65e11}} = sqrt{7.2807e8} ≈ 26,980 ) m/sThese results are consistent with my earlier calculations. So, I think I'm confident that the velocities are approximately 32,940 m/s and 26,980 m/s.Wait, but let me compute the exact values using the formula:( v_p = sqrt{frac{G M (1 + e)}{a (1 - e)}} )Plugging in the numbers:( v_p = sqrt{frac{1.3348e20 times 1.1}{1.5e11 times 0.9}} )Compute numerator: 1.3348e20 * 1.1 = 1.46828e20Denominator: 1.5e11 * 0.9 = 1.35e11So, ( v_p = sqrt{frac{1.46828e20}{1.35e11}} = sqrt{1.0876e9} )Compute sqrt(1.0876e9):Since 32,970^2 = approx 1.087e9, as 32,970^2 = (3.297e4)^2 = 10.87e8 = 1.087e9.So, yes, 32,970 m/s.Similarly, for ( v_a ):( v_a = sqrt{frac{1.3348e20 times 0.9}{1.5e11 times 1.1}} = sqrt{frac{1.20132e20}{1.65e11}} = sqrt{7.2807e8} )sqrt(7.2807e8) = approx 26,980 m/s, since 26,980^2 = approx 7.28e8.Yes, that's correct.So, to summarize:1. The orbital period ( T ) is given by ( T = 2pi sqrt{frac{a^3}{G M}} ).2. The velocities at perihelion and aphelion are approximately 32,940 m/s and 26,980 m/s, respectively.Wait, but let me compute the exact values more precisely.For ( v_p ):Compute ( frac{1.46828e20}{1.35e11} = frac{1.46828}{1.35} times 10^{9} ≈ 1.0876 times 10^9 )sqrt(1.0876e9) = sqrt(1.0876) * 10^4.5 ≈ 1.0428 * 31622.7766 ≈ 32,970 m/s.Similarly, for ( v_a ):Compute ( frac{1.20132e20}{1.65e11} = frac{1.20132}{1.65} times 10^{9} ≈ 0.72807 times 10^9 )sqrt(0.72807e9) = sqrt(0.72807) * 10^4.5 ≈ 0.8533 * 31622.7766 ≈ 26,980 m/s.So, rounding to four significant figures, since the given values have two significant figures for ( a ) and ( e ), but ( M ) has two as well. Wait, actually, ( a = 1.5e11 ) is two significant figures, ( e = 0.1 ) is one, and ( M = 2e30 ) is one. So, perhaps we should round to one significant figure? But that seems too rough.Alternatively, maybe we can keep two significant figures, as ( a ) is given as 1.5e11, which is two, and ( M ) is 2e30, which is one, but perhaps the constants are known more precisely.Wait, but in any case, the velocities are approximately 33,000 m/s and 27,000 m/s.Alternatively, let me compute them more precisely.Compute ( v_p ):sqrt(1.0876e9) = sqrt(1,087,600,000) ≈ 32,970 m/s.Similarly, sqrt(728,070,000) ≈ 26,980 m/s.So, I think these are accurate enough.Final Answer1. The orbital period ( T ) is given by ( boxed{T = 2pi sqrt{frac{a^3}{G M}}} ).2. The velocity at perihelion is approximately ( boxed{33000 , text{m/s}} ) and at aphelion is approximately ( boxed{27000 , text{m/s}} ).</think>

🧠:<think>Okay, so I have this problem about a software development team lead managing five novice programmers. Each programmer has tasks categorized as Easy, Medium, and Hard, with point values 1, 3, and 5 respectively. The first part is about finding possible combinations of tasks for each programmer given their total points. The second part is about determining the weights for a feedback score based on given feedback scores and a ratio for the weights.Starting with part 1. Each programmer has a total point value: P1=30, P2=27, P3=32, P4=29, P5=31. Each programmer's total points are calculated as P_i = e_i + 3m_i + 5h_i, where e_i, m_i, h_i are the number of Easy, Medium, and Hard tasks respectively.So, for each programmer, I need to find all possible non-negative integer solutions (e_i, m_i, h_i) such that e_i + 3m_i + 5h_i = P_i.Let me think about how to approach this. Since the points are 1, 3, and 5, it's similar to solving a Diophantine equation for each programmer. The variables e_i, m_i, h_i must be non-negative integers.I can approach this by fixing h_i first because it has the highest coefficient (5), then m_i, and then e_i will be determined accordingly.Let me take the first programmer, P1=30.So, 30 = e1 + 3m1 + 5h1.I can express this as e1 = 30 - 3m1 - 5h1.Since e1 must be non-negative, 30 - 3m1 - 5h1 ≥ 0.So, 3m1 + 5h1 ≤ 30.Similarly, for each h1, m1 can vary such that 3m1 ≤ 30 - 5h1.So, h1 can range from 0 up to floor(30/5)=6.Similarly, for each h1, m1 can range from 0 up to floor((30 -5h1)/3).So, let's compute possible h1 and m1:h1=0: 3m1 ≤30 => m1=0 to 10h1=1: 3m1 ≤25 => m1=0 to 8 (since 25/3≈8.33)h1=2: 3m1 ≤20 => m1=0 to 6h1=3: 3m1 ≤15 => m1=0 to 5h1=4: 3m1 ≤10 => m1=0 to 3 (since 10/3≈3.33)h1=5: 3m1 ≤5 => m1=0 to 1 (since 5/3≈1.66)h1=6: 3m1 ≤0 => m1=0So, for each h1, m1 can take certain values, and e1 is determined.But since the problem says each programmer is assigned a different combination of tasks. So, each programmer has a unique (e_i, m_i, h_i). So, for each P_i, we need to find all possible (e, m, h) such that e + 3m +5h = P_i, and then assign each programmer a unique combination.But wait, the problem says "each programmer is assigned a different combination of tasks from these categories." So, each programmer has a unique combination, but the total points are given. So, perhaps each programmer's (e, m, h) is unique, but the total points are as given.So, for each programmer, we need to find possible (e, m, h) such that e + 3m +5h = P_i, and all five programmers have distinct (e, m, h) vectors.But the problem is asking for possible combinations, so maybe there are multiple solutions, but we need to find one possible set of combinations.Alternatively, perhaps the problem is just to find possible (e, m, h) for each P_i, regardless of uniqueness across programmers. Hmm.Wait, the question says: "find the possible combinations of (e_i, m_i, h_i) for each programmer." So, perhaps for each P_i, list all possible (e, m, h) that satisfy e + 3m +5h = P_i.But that might be a lot, especially for P=30. So, maybe the problem expects us to find one possible combination for each programmer, such that all are unique.Alternatively, perhaps the problem is just to find possible combinations for each, regardless of uniqueness.But given that the team lead uses a feedback mechanism, and each programmer has a different combination, it's likely that each programmer has a unique combination, so we need to assign each programmer a unique (e, m, h) such that their total points are as given.So, perhaps the task is to find one possible set of combinations where each programmer has a unique (e, m, h) and their total points match the given P_i.So, I need to find five distinct (e, m, h) vectors such that their total points are 30, 27, 32, 29, 31 respectively.Let me try to find such combinations.Starting with P3=32, which is the highest. Let's see what's the maximum h_i can be.h_i can be up to floor(32/5)=6.If h=6, then 5*6=30, so e + 3m = 2. So, possible (e, m) are (2,0) or (0, 2/3), but m must be integer, so only (2,0). So, (e, m, h)=(2,0,6).Alternatively, h=5: 5*5=25, so e +3m=7. Possible m=0: e=7; m=1: e=4; m=2: e=1. So, (7,0,5), (4,1,5), (1,2,5).Similarly, h=4: 20, so e +3m=12. m can be 0-4.So, (12,0,4), (9,1,4), (6,2,4), (3,3,4), (0,4,4).h=3: 15, e +3m=17. m=0-5.(17,0,3), (14,1,3), (11,2,3), (8,3,3), (5,4,3), (2,5,3).h=2: 10, e +3m=22. m=0-7.(22,0,2), (19,1,2), (16,2,2), (13,3,2), (10,4,2), (7,5,2), (4,6,2), (1,7,2).h=1: 5, e +3m=27. m=0-9.(27,0,1), (24,1,1), (21,2,1), (18,3,1), (15,4,1), (12,5,1), (9,6,1), (6,7,1), (3,8,1), (0,9,1).h=0: e +3m=32. m=0-10.(32,0,0), (29,1,0), ..., (2,10,0).So, for P3=32, there are many possible combinations. Let's pick one that might be unique and not interfere with other P_i.Similarly, for P1=30, P2=27, P3=32, P4=29, P5=31.I think it's better to pick combinations where h_i is as high as possible to minimize e_i, but let's see.Alternatively, maybe the team lead assigned tasks in a way that each programmer has a different number of hard tasks, medium tasks, etc.But perhaps the simplest way is to assign each programmer a unique combination where h_i is unique.So, let's try to assign h_i from 0 to 4 or something.Wait, but P3=32 is the highest, so maybe h_i=6 for P3.Similarly, P5=31: h_i=6 would give 5*6=30, so e +3m=1. So, (1,0,6) or (0, 1/3,6), but m must be integer, so only (1,0,6). But P3 already has h=6, so maybe P5 can't have h=6. So, perhaps P3 has h=6, and P5 has h=5.Wait, let's see:If P3=32, h=6: e +3m=2. So, (2,0,6). That's one possibility.Then P5=31: h=6 would require e +3m=1, which is only (1,0,6). But h=6 is already used by P3, so maybe P5 can have h=5.h=5: 25, so e +3m=6. Possible (6,0,5), (3,1,5), (0,2,5).Similarly, P1=30: h=6: e +3m=0, which is (0,0,6). But that's only if h=6, but P3 already has h=6, so P1 can't have h=6. So, P1 can have h=5: 25, e +3m=5. So, (5,0,5), (2,1,5), (0, 5/3,5) which is invalid. So, (5,0,5) or (2,1,5).Alternatively, h=4: 20, e +3m=10. So, (10,0,4), (7,1,4), (4,2,4), (1,3,4).Similarly, for P2=27: h=5: 25, e +3m=2: (2,0,5), (0, 2/3,5) invalid. So, (2,0,5). Or h=4: 20, e +3m=7: (7,0,4), (4,1,4), (1,2,4).P4=29: h=5: 25, e +3m=4: (4,0,5), (1,1,5). Or h=4: 20, e +3m=9: (9,0,4), (6,1,4), (3,2,4), (0,3,4).P5=31: h=6: e +3m=1: (1,0,6). But if P3 has h=6, then P5 can't have h=6. So, P5 can have h=5: e +3m=6: (6,0,5), (3,1,5), (0,2,5).So, let's try to assign h_i as follows:P3=32: h=6, so (2,0,6)P5=31: h=5, let's pick (6,0,5)P1=30: h=5, but P5 already has h=5, so P1 can have h=4: 20, e +3m=10. Let's pick (10,0,4)P2=27: h=4: 20, e +3m=7. Let's pick (7,0,4)P4=29: h=4: 20, e +3m=9. Let's pick (9,0,4)Wait, but P1, P2, P4 all have h=4, which might not be unique. The problem says each programmer is assigned a different combination, but it doesn't specify that h_i must be unique. So, maybe it's okay if h_i are the same, as long as the entire vector (e, m, h) is unique.So, let's check:P3: (2,0,6)P5: (6,0,5)P1: (10,0,4)P2: (7,0,4)P4: (9,0,4)Wait, but P1, P2, P4 all have m=0 and h=4, but different e. So, their vectors are (10,0,4), (7,0,4), (9,0,4). These are distinct because e is different. So, that's acceptable.But let's check if these combinations are valid:For P1=30: 10 + 0 + 20 = 30. Yes.P2=27: 7 + 0 + 20 =27. Yes.P3=32: 2 +0 +30=32. Yes.P4=29:9 +0 +20=29. Yes.P5=31:6 +0 +25=31. Yes.So, that works. But let's see if we can have more varied combinations, maybe with some m_i>0.Alternatively, let's try to assign each programmer a unique h_i.So, h_i can be 6,5,4,3,2 for the five programmers.So, P3=32: h=6, e +3m=2. So, (2,0,6)P5=31: h=5, e +3m=6. Let's pick (3,1,5)P1=30: h=4, e +3m=10. Let's pick (1,3,4) because 1 + 9 + 20=30.Wait, e=1, m=3, h=4: 1 + 9 + 20=30. Yes.P2=27: h=3, e +3m=12. Let's pick (3,3,3): 3 +9 +15=27.P4=29: h=2, e +3m=19. Let's pick (1,6,2): 1 +18 +10=29.So, let's check all:P1: (1,3,4) =1 +9 +20=30P2: (3,3,3)=3 +9 +15=27P3: (2,0,6)=2 +0 +30=32P4: (1,6,2)=1 +18 +10=29P5: (3,1,5)=3 +3 +25=31Yes, all totals match. Also, each programmer has a unique combination:(1,3,4), (3,3,3), (2,0,6), (1,6,2), (3,1,5). All are distinct.So, that's a possible set of combinations.Alternatively, there are many other possibilities, but this seems to satisfy the conditions.Now, moving to part 2.The feedback score F_i is given by F_i = α e_i + β m_i + γ h_i, where α:β:γ=1:2:3, and α + β + γ=10.We need to find α, β, γ.Given the ratio α:β:γ=1:2:3, let's denote α=1k, β=2k, γ=3k for some k.Then, α + β + γ=1k +2k +3k=6k=10.So, 6k=10 => k=10/6=5/3≈1.6667.Thus, α=5/3, β=10/3, γ=5.But let's check if this works with the given feedback scores.Given:F1=40, F2=35, F3=45, F4=38, F5=42.Using the combinations we found earlier:P1: (1,3,4) => F1= α*1 + β*3 + γ*4= (5/3)*1 + (10/3)*3 +5*4= 5/3 +10 +20= 5/3 +30=31.666..., but F1 is given as 40. So, this doesn't match.Hmm, that's a problem. So, maybe my assumption about the combinations is incorrect, or perhaps I need to adjust the weights.Wait, but the weights are determined based on the feedback scores, so perhaps the combinations I chose don't align with the feedback scores given. So, maybe I need to find the correct weights that satisfy all feedback scores.Alternatively, perhaps I need to use the feedback scores to solve for α, β, γ.Given that α:β:γ=1:2:3, so β=2α, γ=3α.And α + β + γ=10 => α +2α +3α=6α=10 => α=10/6=5/3≈1.6667.So, α=5/3, β=10/3, γ=5.But then, using the combinations I found earlier, the feedback scores don't match. So, perhaps the combinations I chose are not the ones used by the team lead, or perhaps I need to find different combinations that satisfy both the total points and the feedback scores with these weights.Alternatively, maybe the feedback scores are calculated with these weights, so we can use them to find the weights, but since the ratio is given, we can directly compute α, β, γ as above.Wait, but the problem says: "the weights are such that α + β + γ =10 with α : β : γ =1 : 2 :3, determine the values of α, β, γ."So, regardless of the feedback scores, since the ratio and sum are given, we can directly compute α, β, γ.So, as above, α=5/3, β=10/3, γ=5.But let's check if these weights can produce the given feedback scores with the combinations we found.For P1: (1,3,4): F1=5/3*1 +10/3*3 +5*4=5/3 +10 +20=5/3 +30≈31.666, but F1=40. So, discrepancy.Similarly, for P3: (2,0,6): F3=5/3*2 +10/3*0 +5*6=10/3 +0 +30≈10/3 +30≈33.333, but F3=45. So, not matching.Thus, either the combinations I chose are incorrect, or the weights are different.But wait, the problem says that the weights are such that α:β:γ=1:2:3 and α + β + γ=10. So, regardless of the feedback scores, we can compute α, β, γ as 5/3, 10/3, 5.But then, why are the feedback scores given? Maybe to verify that the weights are correct, but in this case, with the combinations I chose, they don't match. So, perhaps the combinations I chose are not the ones used, and the feedback scores are consistent with the weights.Alternatively, perhaps the feedback scores are given to help determine the weights, but since the ratio is given, we can compute them directly.Wait, but if the ratio is given, then the weights are fixed as α=5/3, β=10/3, γ=5, regardless of the feedback scores. So, perhaps the feedback scores are just additional information, but the weights are determined solely by the ratio and sum.So, perhaps the answer is α=5/3, β=10/3, γ=5.But let's see if these weights can produce the feedback scores with some combinations.Given F_i= α e_i + β m_i + γ h_i.Given that, for each programmer, F_i is given, and P_i is given, which is e_i +3m_i +5h_i.So, we have two equations for each programmer:1. e_i +3m_i +5h_i = P_i2. (5/3)e_i + (10/3)m_i +5h_i = F_iLet me write these as:Equation 1: e +3m +5h = PEquation 2: (5/3)e + (10/3)m +5h = FLet me multiply Equation 2 by 3 to eliminate denominators:5e +10m +15h = 3FNow, subtract Equation 1 multiplied by 5:Equation 1 *5: 5e +15m +25h =5PSubtract from Equation 2 *3:(5e +10m +15h) - (5e +15m +25h) = 3F -5PSimplify:-5m -10h =3F -5PDivide both sides by -5:m +2h = (5P -3F)/5So, m +2h = P - (3F)/5But m and h are integers, so (5P -3F) must be divisible by 5.Let me check for each programmer:For P1: P=30, F=405P -3F=150 -120=3030/5=6So, m +2h=6Similarly, for P2: P=27, F=355*27 -3*35=135 -105=3030/5=6m +2h=6P3: P=32, F=455*32 -3*45=160 -135=2525/5=5m +2h=5P4: P=29, F=385*29 -3*38=145 -114=3131/5=6.2, which is not an integer. Hmm, problem.Wait, 5P -3F=145 -114=31, which is not divisible by 5. So, m +2h=31/5=6.2, which is not possible since m and h are integers.Similarly, P5: P=31, F=425*31 -3*42=155 -126=2929/5=5.8, not integer.So, for P4 and P5, m +2h is not integer, which is impossible. Therefore, the weights I calculated cannot produce the given feedback scores with integer m and h.This suggests that either the weights are different, or the combinations are different.But wait, the problem states that the weights are such that α + β + γ=10 and α:β:γ=1:2:3. So, the weights are fixed as α=5/3, β=10/3, γ=5.But then, the feedback scores would require that for each programmer, m +2h=(5P -3F)/5 is integer.But for P4 and P5, it's not. So, perhaps the feedback scores are incorrect, or the combinations are different.Alternatively, maybe I made a mistake in the approach.Wait, perhaps instead of using the combinations I found earlier, I need to find combinations that satisfy both the total points and the feedback scores with the given weights.So, let's try to find (e, m, h) for each programmer such that:1. e +3m +5h = P_i2. (5/3)e + (10/3)m +5h = F_iLet me rewrite equation 2:Multiply by 3: 5e +10m +15h =3F_iFrom equation 1: e = P_i -3m -5hSubstitute into equation 2:5(P_i -3m -5h) +10m +15h =3F_i5P_i -15m -25h +10m +15h =3F_i5P_i -5m -10h =3F_iDivide both sides by 5:P_i -m -2h = (3F_i)/5So, P_i -m -2h must be equal to (3F_i)/5.Since m and h are integers, (3F_i)/5 must be integer.Let's check for each programmer:P1: F1=40, 3*40=120, 120/5=24. So, P1 -m -2h=24. Since P1=30, 30 -m -2h=24 => m +2h=6.Similarly, P2: F2=35, 3*35=105, 105/5=21. P2=27, so 27 -m -2h=21 => m +2h=6.P3: F3=45, 3*45=135, 135/5=27. P3=32, so 32 -m -2h=27 => m +2h=5.P4: F4=38, 3*38=114, 114/5=22.8, which is not integer. So, problem.P5: F5=42, 3*42=126, 126/5=25.2, not integer.So, again, P4 and P5 have non-integer results, which is impossible. Therefore, the given feedback scores are inconsistent with the weights derived from the ratio and sum.This suggests that either the problem has a mistake, or perhaps I misunderstood the ratio.Wait, the ratio is α:β:γ=1:2:3, so α=1k, β=2k, γ=3k, and α + β + γ=10 =>6k=10 =>k=10/6=5/3.So, α=5/3, β=10/3, γ=5.But with these weights, the feedback scores for P4 and P5 cannot be integers, which contradicts the given feedback scores.Therefore, perhaps the ratio is not 1:2:3, but 1:2:3 in terms of their weights, but not necessarily in terms of their coefficients.Wait, no, the ratio is given as α:β:γ=1:2:3, so it's 1:2:3.Alternatively, maybe the ratio is in terms of their contributions, but perhaps the weights are integers.Wait, if α:β:γ=1:2:3, and α + β + γ=10, then the only integer solution is α=1, β=2, γ=3, but 1+2+3=6≠10. So, scaling up, 10/6=5/3, so α=5/3, β=10/3, γ=5.But then, as above, feedback scores for P4 and P5 are not integers.Alternatively, perhaps the ratio is 1:2:3, but the weights are integers. So, let's see:Let α=1k, β=2k, γ=3k, with k integer.Then, α + β + γ=6k=10. But 10 is not divisible by 6, so no integer solution.Thus, the weights must be fractions.Therefore, the only possible weights are α=5/3, β=10/3, γ=5.But then, the feedback scores for P4 and P5 cannot be integers, which contradicts the given feedback scores.This suggests that either the problem has an inconsistency, or perhaps I made a mistake in the approach.Alternatively, perhaps the ratio is not 1:2:3, but 1:2:3 in terms of their weights, but the sum is 10. So, perhaps the weights are 1x, 2x, 3x, and 1x +2x +3x=10 =>6x=10 =>x=10/6=5/3.Thus, α=5/3, β=10/3, γ=5.But as above, this leads to non-integer feedback scores for P4 and P5.Therefore, perhaps the problem expects us to ignore the feedback scores and just compute the weights based on the ratio and sum, regardless of whether they produce integer feedback scores.So, the answer would be α=5/3, β=10/3, γ=5.Alternatively, perhaps the feedback scores are given to help determine the weights, but since the ratio is given, we can compute them directly.So, in conclusion, the weights are α=5/3, β=10/3, γ=5.</think>