answer: Solution# Part 1: Determine the time ( t ) at which the mental state first returns to its initial state.The mental state ( M(t) ) is given by:[ M(t) = e^{i(2pi t + theta)} ]For the mental state to return to its initial state, we need:[ M(t) = M(0) ]Evaluating ( M(0) ):[ M(0) = e^{itheta} ]We need to find ( t ) such that:[ e^{i(2pi t + theta)} = e^{itheta} ]This equality holds when:[ 2pi t + theta = theta + 2pi k ]for some integer ( k ).Simplifying, we get:[ 2pi t = 2pi k ][ t = k ]The smallest positive ( t ) is when ( k = 1 ):[ t = 1 ]So, the mental state first returns to its initial state at ( t = 1 ).# Part 2: Find the physical health ( P(t) ) after the mental state undergoes one complete cycle.The physical health ( P(t) ) is given by:[ P(t) = int_0^t sin(M(tau)) , dtau ]We need to evaluate ( P(t) ) after one complete cycle, which is at ( t = 1 ).First, let's express ( sin(M(tau)) ):[ M(tau) = e^{i(2pi tau + theta)} ]Since ( e^{iphi} = cos(phi) + isin(phi) ), we have:[ M(tau) = cos(2pi tau + theta) + isin(2pi tau + theta) ]So,[ sin(M(tau)) = sinleft( cos(2pi tau + theta) + isin(2pi tau + theta) right) ]Evaluating ( sin(M(tau)) ) directly seems complicated. Instead, we focus on the integral over one period. Given the periodicity of ( M(t) ) and the symmetry of sine over one period, it turns out that the integral of the sine of a complex exponential over one period is zero, due to the orthogonality of sine and cosine functions integrated over their period.Thus, we have:[ P(1) = int_0^1 sin(M(tau)) , dtau = 0 ]Final Answer: The final answer is ( P(1) = 0 ). I hope it is correct.

answer:Let's solve the problem step-by-step. Part 1: Population Proportions After the First CampaignInitially, the proportions of the population in groups A, B, and C are given by (p_A), (p_B), and (p_C) respectively, with the constraint (p_A + p_B + p_C = 1).After the first campaign:- The proportion of Group A increases by 10%, so the new proportion of Group A is (p_A' = p_A + 0.10p_A = 1.10p_A).- The proportion of Group B decreases by 5%, so the new proportion of Group B is (p_B' = p_B - 0.05p_B = 0.95p_B).Since the total proportion must still sum to 1, the new proportion of Group C can be found by:[ p_C' = 1 - p_A' - p_B' ]Substituting the new values of (p_A') and (p_B'):[ p_C' = 1 - 1.10p_A - 0.95p_B ]Therefore, the new proportions are:[ p_A' = 1.10p_A ][ p_B' = 0.95p_B ][ p_C' = 1 - 1.10p_A - 0.95p_B ] Part 2: Population Proportions After the Second CampaignThe transformation matrix (T) is given as:[T = begin{pmatrix}0.90 & 0.05 & 0.05 0.05 & 0.85 & 0.10 0.05 & 0.10 & 0.85end{pmatrix}]The population proportions after the first campaign can be represented by the vector:[mathbf{P_1} = begin{pmatrix} p_A' p_B' p_C' end{pmatrix} = begin{pmatrix} 1.10p_A 0.95p_B 1 - 1.10p_A - 0.95p_B end{pmatrix}]To find the population proportions after the second campaign, we compute (mathbf{P_2} = T mathbf{P_1}):[mathbf{P_2} = begin{pmatrix}0.90 & 0.05 & 0.05 0.05 & 0.85 & 0.10 0.05 & 0.10 & 0.85end{pmatrix}begin{pmatrix} 1.10p_A 0.95p_B 1 - 1.10p_A - 0.95p_B end{pmatrix}]Let's compute each element of (mathbf{P_2}):1. For the first element ((p_A'')):[p_A'' = 0.90(1.10p_A) + 0.05(0.95p_B) + 0.05(1 - 1.10p_A - 0.95p_B)]Simplifying:[p_A'' = 0.99p_A + 0.0475p_B + 0.05 - 0.055p_A - 0.0475p_B][p_A'' = 0.935p_A + 0.05]2. For the second element ((p_B'')):[p_B'' = 0.05(1.10p_A) + 0.85(0.95p_B) + 0.10(1 - 1.10p_A - 0.95p_B)]Simplifying:[p_B'' = 0.055p_A + 0.8075p_B + 0.10 - 0.11p_A - 0.095p_B][p_B'' = -0.055p_A + 0.7125p_B + 0.10]3. For the third element ((p_C'')):[p_C'' = 0.05(1.10p_A) + 0.10(0.95p_B) + 0.85(1 - 1.10p_A - 0.95p_B)]Simplifying:[p_C'' = 0.055p_A + 0.095p_B + 0.85 - 0.935p_A - 0.8075p_B][p_C'' = -0.88p_A - 0.7125p_B + 0.85] Final Answer:The final proportions of the population after the second campaign are:[mathbf{P_2} = begin{pmatrix} 0.935p_A + 0.05 -0.055p_A + 0.7125p_B + 0.10 -0.88p_A - 0.7125p_B + 0.85 end{pmatrix}]Final Answer: The final answer is (mathbf{P_2} = begin{pmatrix} 0.935p_A + 0.05 -0.055p_A + 0.7125p_B + 0.10 -0.88p_A - 0.7125p_B + 0.85 end{pmatrix}). I hope it is correct.

answer:To solve this problem, we need to tackle it in two parts: finding the critical points of the function and maximizing the function under a constraint. Part 1: Finding Critical PointsThe function is given by:[ f(x, y, z) = x^2y + yz^3 - e^{x+y} ]Step 1: Compute the partial derivatives1. Partial derivative with respect to ( x ): [ f_x = frac{partial f}{partial x} = 2xy - e^{x+y} ]2. Partial derivative with respect to ( y ): [ f_y = frac{partial f}{partial y} = x^2 + z^3 - e^{x+y} ]3. Partial derivative with respect to ( z ): [ f_z = frac{partial f}{partial z} = 3yz^2 ]Step 2: Solve the system of equations by setting the partial derivatives to zero1. ( 2xy - e^{x+y} = 0 )2. ( x^2 + z^3 - e^{x+y} = 0 )3. ( 3yz^2 = 0 )From the third equation, ( 3yz^2 = 0 ), we have two possible cases:- ( y = 0 )- ( z = 0 )Case 1: ( y = 0 )Substitute ( y = 0 ) into the first two equations:- ( 2x cdot 0 - e^{x+0} = 0 ) implies ( e^x = 0 ), which is impossible.- ( x^2 + z^3 - e^{x+0} = 0 ) simplifies to ( x^2 + z^3 - e^x = 0 ).Since ( e^x > 0 ) for all ( x ), there are no real solutions in this case.Case 2: ( z = 0 )Substitute ( z = 0 ) into the first two equations:- ( 2xy - e^{x+y} = 0 ) becomes ( 2xy = e^{x+y} ).- ( x^2 + 0^3 - e^{x+y} = 0 ) simplifies to ( x^2 = e^{x+y} ).Now we have:1. ( 2xy = e^{x+y} )2. ( x^2 = e^{x+y} )From these, we equate:[ 2xy = x^2 ]This gives us:- If ( x neq 0 ), then ( 2y = x ) or ( y = frac{x}{2} ).Now substitute ( y = frac{x}{2} ) into ( x^2 = e^{x+y} ):[ x^2 = e^{x+frac{x}{2}} = e^{frac{3x}{2}} ]This equation is transcendental and typically solved numerically. For simplicity here, we note:- If ( x = 0 ), then ( y = 0 ) and ( x^2 = e^0 = 1 ), which does not hold.Thus, the critical points must be determined numerically or graphically. Part 2: Maximizing the Function Subject to a ConstraintGiven the constraint:[ x + y + z = 10 ]with proportions ( x : y : z = 1 : 2 : 3 ), we have:[ x = k, , y = 2k, , z = 3k ]Substitute into the constraint:[ k + 2k + 3k = 10 ][ 6k = 10 ][ k = frac{10}{6} = frac{5}{3} ]Thus, the values of ( x, y, z ) are:- ( x = frac{5}{3} )- ( y = frac{10}{3} )- ( z = 5 ) Final Step: Evaluate the function at these values:[ fleft(frac{5}{3}, frac{10}{3}, 5right) = left(frac{5}{3}right)^2 left(frac{10}{3}right) + left(frac{10}{3}right) cdot 5^3 - e^{frac{5}{3} + frac{10}{3}} ]Simplify:[ fleft(frac{5}{3}, frac{10}{3}, 5right) = frac{25}{9} cdot frac{10}{3} + frac{10}{3} cdot 125 - e^5 ][ = frac{250}{27} + frac{1250}{3} - e^5 ]This calculation can be further simplified numerically.Final Answer:The final answer is: The optimal amounts are ( x = frac{5}{3} ), ( y = frac{10}{3} ), ( z = 5 ) with the flavor profile ( fleft(frac{5}{3}, frac{10}{3}, 5right) ) calculated as needed. I hope it is correct.

answer:To solve the given problem, we need to carry out the following steps: Part 1: Calculate the total energy consumed by the refrigerator in a 24-hour periodThe energy consumption of the refrigerator is given by ( E_r(t) = 0.5 + 0.3sin(pi t / 12) ).To find the total energy consumed over 24 hours, we need to integrate this function from ( t = 0 ) to ( t = 24 ).[int_0^{24} E_r(t) , dt = int_0^{24} left( 0.5 + 0.3sinleft(frac{pi t}{12}right) right) , dt]This integral can be split into two separate integrals:[= int_0^{24} 0.5 , dt + int_0^{24} 0.3sinleft(frac{pi t}{12}right) , dt]1. Calculate the first integral:[int_0^{24} 0.5 , dt = 0.5 times 24 = 12]2. Calculate the second integral:Let ( u = frac{pi t}{12} ), then ( du = frac{pi}{12} , dt ) or ( dt = frac{12}{pi} , du ).When ( t = 0 ), ( u = 0 ). When ( t = 24 ), ( u = 2pi ).[int_0^{24} 0.3sinleft(frac{pi t}{12}right) , dt = 0.3 int_0^{2pi} sin(u) cdot frac{12}{pi} , du][= frac{3.6}{pi} int_0^{2pi} sin(u) , du]The integral of (sin(u)) from (0) to (2pi) is zero because the areas above and below the x-axis cancel each other out over one full period:[int_0^{2pi} sin(u) , du = 0]Thus, the second integral is (0).Adding both integrals gives the total energy consumption of the refrigerator over 24 hours:[12 + 0 = 12 , text{kWh}]The average energy consumption of the refrigerator over 24 hours is:[frac{12}{24} = 0.5 , text{kWh}] Part 2: Determine the time intervals when the heating system's energy consumption exceeds the average energy consumption of the refrigeratorThe energy consumption of the heating system is modeled by the piecewise function:[E_h(t) = begin{cases} 2, & text{if } 0 leq t < 8 1.5, & text{if } 8 leq t < 16 3 - 0.1t, & text{if } 16 leq t leq 24 end{cases}]We compare each piece with the average refrigerator energy consumption of ( 0.5 , text{kWh} ):- For ( 0 leq t < 8 ): ( E_h(t) = 2 ), which is greater than ( 0.5 ).- For ( 8 leq t < 16 ): ( E_h(t) = 1.5 ), which is greater than ( 0.5 ).- For ( 16 leq t leq 24 ): ( E_h(t) = 3 - 0.1t ). We solve ( 3 - 0.1t > 0.5 ): [ 3 - 0.1t > 0.5 implies 2.5 > 0.1t implies 25 > t ] Since ( t leq 24 ), the inequality holds for the entire interval ( 16 leq t leq 24 ).Thus, the heating system's energy consumption exceeds the average refrigerator's consumption over the entire 24-hour period.Final Answer: The final answer is the heating system's energy consumption exceeds the average refrigerator consumption throughout the entire day (0 ≤ t ≤ 24). I hope it is correct.

answer:Let's solve the problem step-by-step: Step 1: Formulate the System of EquationsGiven:- The total wage budget for all employees is 1,000,000.- The sum of all productivity scores is 5000.- The sum of all satisfaction coefficients is 5000.The wage for employee ( i ) is given by:[ W_i = k cdot P_i^{alpha} cdot S_i^{beta} ]We need to find the sum of the wages for all employees:[ sum_{i=1}^{100} W_i = 1,000,000 ]Substituting ( W_i ) into the equation, we get:[ sum_{i=1}^{100} k cdot P_i^{alpha} cdot S_i^{beta} = 1,000,000 ]Since ( k ) is a constant, we can factor it out:[ k cdot sum_{i=1}^{100} P_i^{alpha} cdot S_i^{beta} = 1,000,000 ]Given ( alpha = 0.5 ) and ( beta = 0.5 ), the above equation becomes:[ k cdot sum_{i=1}^{100} P_i^{0.5} cdot S_i^{0.5} = 1,000,000 ] Step 2: Calculate the Value of ( k )We are given that the sum of all productivity scores ( sum_{i=1}^{100} P_i = 5000 ) and the sum of all satisfaction coefficients ( sum_{i=1}^{100} S_i = 5000 ).To simplify the calculation, let's assume that productivity scores and satisfaction coefficients are uniformly distributed among employees. Therefore, each employee ( i ) has an average productivity score ( bar{P} ) and an average satisfaction coefficient ( bar{S} ):[ bar{P} = frac{5000}{100} = 50 ][ bar{S} = frac{5000}{100} = 50 ]Then,[ sum_{i=1}^{100} P_i^{0.5} cdot S_i^{0.5} approx 100 cdot (50^{0.5} cdot 50^{0.5}) ]Since ( 50^{0.5} cdot 50^{0.5} = 50 ):[ sum_{i=1}^{100} 50 = 100 cdot 50 = 5000 ]Thus, we have:[ k cdot 5000 = 1,000,000 ]Solving for ( k ):[ k = frac{1,000,000}{5000} = 200 ] Step 3: Determine the Wage for a Specific EmployeeGiven:- Productivity score ( P = 80 )- Satisfaction coefficient ( S = 90 )The wage ( W ) for this employee is:[ W = k cdot P^{0.5} cdot S^{0.5} ]Substitute the values:[ W = 200 cdot 80^{0.5} cdot 90^{0.5} ]Calculate ( 80^{0.5} ) and ( 90^{0.5} ):[ 80^{0.5} = sqrt{80} approx 8.944 ][ 90^{0.5} = sqrt{90} approx 9.487 ]So,[ W = 200 cdot 8.944 cdot 9.487 approx 200 cdot 84.865 approx 16973 ]Final Answer: The final answer is 16973. I hope it is correct.

answer:Let's tackle each part of the problem step-by-step. Part 1: Probability that all version numbers satisfy ( V_i > frac{1}{n} )The version numbers ( V_1, V_2, ldots, V_n ) are independent and identically distributed random variables, each uniformly distributed over the interval ([0, 1)). We need to find the probability that all these version numbers are greater than (frac{1}{n}).For a single version number ( V_i ), the probability that ( V_i > frac{1}{n} ) is:[P(V_i > frac{1}{n}) = 1 - P(V_i leq frac{1}{n}) = 1 - frac{1}{n} = frac{n-1}{n}]Since the version numbers are independent, the probability that all ( n ) version numbers satisfy ( V_i > frac{1}{n} ) is:[P(V_1 > frac{1}{n}, V_2 > frac{1}{n}, ldots, V_n > frac{1}{n}) = left( frac{n-1}{n} right)^n] Part 2: Expected number of errors and varianceLet ( X_i ) be a random variable indicating whether the ( i )-th transaction results in an error. ( X_i = 1 ) if there is an error, and ( X_i = 0 ) if there is no error. The probability of an error for each transaction is ( p ). Therefore, ( X_i ) follows a Bernoulli distribution with parameter ( p ).The total number of erroneous transactions, ( X ), is the sum of these individual Bernoulli random variables:[X = X_1 + X_2 + cdots + X_n]Expected Number of Errors:The expected value of a sum of independent random variables is the sum of their expected values. So, the expected number of errors is:[E[X] = E[X_1] + E[X_2] + cdots + E[X_n] = n cdot p]Variance of the Number of Errors:The variance of a sum of independent random variables is the sum of their variances. The variance of a Bernoulli random variable ( X_i ) with parameter ( p ) is:[text{Var}(X_i) = p(1-p)]Thus, the variance of the total number of errors is:[text{Var}(X) = text{Var}(X_1) + text{Var}(X_2) + cdots + text{Var}(X_n) = n cdot p(1-p)]Final Answer: The final answer is (left(left( frac{n-1}{n} right)^n, n cdot p, n cdot p(1-p)right)). I hope it is correct.