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question:Pete Nelson, the host of Treehouse Masters, decides to build a new, unique treehouse. He wants the treehouse to be supported by three strong trees forming the vertices of an equilateral triangle. The side length of the triangle is 30 feet. 1. Determine the coordinates of the vertices if one vertex is at ((0, 0)) and the triangle is oriented so that one side is parallel to the x-axis. Use these coordinates to calculate the centroid of the triangle, which will be the center of mass where Pete wants to place the central support beam.2. Pete also wants to install a spiral staircase inside the treehouse. The staircase will start from the ground at ((0, 0, 0)) and ascend to a height of 20 feet, completing exactly 3 full rotations around the z-axis. Using the parametric equations for a helix, find the length of the staircase.Note: The parametric equations for a helix are given by (x(t) = R cos(t)), (y(t) = R sin(t)), and (z(t) = frac{h}{2pi}t), where (R) is the radius, (h) is the height, and (t) ranges from 0 to (6pi) for 3 full rotations.
answer:Let's work through the problem step by step.Step 1: Determine the coordinates of the vertices of the equilateral triangle.Given:- One vertex is at ((0, 0)).- The side length of the triangle is 30 feet.- The triangle is oriented so that one side is parallel to the x-axis.For an equilateral triangle with one vertex at ((0, 0)) and the side parallel to the x-axis:- The second vertex will be at ((30, 0)).- The third vertex will be ((x_3, y_3)).To find the third vertex:1. The distance between any two vertices of the equilateral triangle is 30 feet.2. The coordinates of the third vertex can be derived using the properties of equilateral triangles.For the third vertex ((x_3, y_3)):- Let the coordinates be ((15, y_3)) because the third vertex forms an equilateral triangle, and it must be equidistant from both ((0, 0)) and ((30, 0)).- The distance from ((0, 0)) to ((15, y_3)) is 30 feet.Using the distance formula:[sqrt{(15-0)^2 + (y_3-0)^2} = 30][sqrt{225 + y_3^2} = 30][225 + y_3^2 = 900][y_3^2 = 675][y_3 = sqrt{675} = 15sqrt{3}]So, the coordinates of the third vertex are ((15, 15sqrt{3})).The coordinates of the vertices of the equilateral triangle are:- ((0, 0))- ((30, 0))- ((15, 15sqrt{3}))Step 2: Calculate the centroid of the triangle.The centroid (G) of a triangle with vertices ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)) is given by:[G = left(frac{x_1 + x_2 + x_3}{3}, frac{y_1 + y_2 + y_3}{3}right)]Using the vertices ((0, 0)), ((30, 0)), and ((15, 15sqrt{3})):[x_G = frac{0 + 30 + 15}{3} = frac{45}{3} = 15][y_G = frac{0 + 0 + 15sqrt{3}}{3} = frac{15sqrt{3}}{3} = 5sqrt{3}]The centroid of the triangle is ((15, 5sqrt{3})).Step 3: Calculate the length of the spiral staircase.Given:- The staircase ascends from ((0, 0, 0)) to a height of 20 feet.- It completes exactly 3 full rotations around the z-axis.- Parametric equations: (x(t) = R cos(t)), (y(t) = R sin(t)), and (z(t) = frac{h}{2pi}t).For the height (h = 20) feet and 3 full rotations:- (t) ranges from 0 to (6pi).The height equation is:[z(t) = frac{20}{6pi}t = frac{10}{3pi}t]The radius (R) can be chosen to be 15 feet, centered at the centroid since that is a reasonable assumption for a central helix.The parametric equations are:[x(t) = 15 cos(t)][y(t) = 15 sin(t)][z(t) = frac{10}{3pi}t]To find the length of the helix, we use the arc length formula for parametric curves:[L = int_{a}^{b} sqrt{left(frac{dx}{dt}right)^2 + left(frac{dy}{dt}right)^2 + left(frac{dz}{dt}right)^2} , dt]Calculating the derivatives:[frac{dx}{dt} = -15 sin(t)][frac{dy}{dt} = 15 cos(t)][frac{dz}{dt} = frac{10}{3pi}]Substitute these into the arc length formula:[L = int_{0}^{6pi} sqrt{(-15 sin(t))^2 + (15 cos(t))^2 + left(frac{10}{3pi}right)^2} , dt][L = int_{0}^{6pi} sqrt{225 sin^2(t) + 225 cos^2(t) + left(frac{10}{3pi}right)^2} , dt][L = int_{0}^{6pi} sqrt{225 (sin^2(t) + cos^2(t)) + left(frac{10}{3pi}right)^2} , dt][L = int_{0}^{6pi} sqrt{225 + left(frac{10}{3pi}right)^2} , dt][L = int_{0}^{6pi} sqrt{225 + frac{100}{9pi^2}} , dt][L = int_{0}^{6pi} sqrt{225 + frac{100}{9pi^2}} , dt][L = int_{0}^{6pi} sqrt{225 + frac{100}{9pi^2}} , dt][L = sqrt{225 + frac{100}{9pi^2}} int_{0}^{6pi} , dt][L = sqrt{225 + frac{100}{9pi^2}} cdot 6pi][L = 6pi cdot sqrt{225 + frac{100}{9pi^2}}]Simplify the expression inside the square root:[225 + frac{100}{9pi^2} = 225 + frac{100}{9pi^2}]Since (frac{100}{9pi^2}) is relatively small compared to 225, approximate the square root:[sqrt{225 + frac{100}{9pi^2}} approx 15]Thus, the length is approximately:[L approx 6pi times 15 = 90pi]Final Answer: The centroid of the triangle is ((15, 5sqrt{3})). The length of the staircase is approximately (90pi) feet. I hope it is correct.
question:A trauma survivor attends therapy sessions to overcome anxiety and PTSD. The psychologist employs a highly empathetic approach and tracks the progress through a series of quantitative assessments. Let ( T(t) ) represent the trauma survivor's anxiety level as a function of time ( t ) in weeks, where ( T(t) ) follows the differential equation:[ frac{dT}{dt} = -aT + bsin(ct) ]Here, ( a ), ( b ), and ( c ) are positive constants representing various factors of therapy effectiveness and external stressors.1. Given the initial condition ( T(0) = T_0 ), find the general solution of the differential equation ( frac{dT}{dt} = -aT + bsin(ct) ).2. If the trauma survivor's anxiety level is observed to oscillate with a maximum amplitude of ( A ) after a long period, determine the relationship between the constants ( a ), ( b ), and ( c ) in terms of ( A ).
answer:Let's solve the given differential equation step-by-step.Step 1: Solve the Differential EquationThe differential equation given is:[ frac{dT}{dt} = -aT + bsin(ct) ]This is a linear non-homogeneous differential equation. To solve it, we need to find the general solution of the homogeneous part and a particular solution of the non-homogeneous part.Step 1.1: Solve the Homogeneous EquationThe homogeneous part of the equation is:[ frac{dT}{dt} + aT = 0 ]To solve this, we can use the method of separation of variables:[ frac{dT}{T} = -a dt ]Integrating both sides:[ int frac{1}{T} dT = int -a dt ][ ln |T| = -at + C ]Exponentiating both sides to solve for ( T ):[ T = e^{-at + C} = e^C e^{-at} ]Let ( C_1 = e^C ). Thus, the solution to the homogeneous equation is:[ T_h(t) = C_1 e^{-at} ]Step 1.2: Find a Particular SolutionWe need a particular solution ( T_p(t) ) of the non-homogeneous equation. We can try a solution of the form:[ T_p(t) = A sin(ct) + B cos(ct) ]Taking the derivative of ( T_p(t) ):[ frac{dT_p}{dt} = Accos(ct) - Bcsin(ct) ]Substitute ( T_p(t) ) and its derivative into the original differential equation:[ Accos(ct) - Bcsin(ct) = -a(Asin(ct) + Bcos(ct)) + bsin(ct) ]Collecting like terms:[ Accos(ct) + aBcos(ct) = bsin(ct) - aAsin(ct) - Bcsin(ct) ]Separate terms involving ( sin(ct) ) and ( cos(ct) ):[ (Ac + aB)cos(ct) + (b - aA - Bc)sin(ct) = 0 ]This gives us two equations by setting the coefficients of (cos(ct)) and (sin(ct)) to zero:1. ( Ac + aB = 0 )2. ( b - aA - Bc = 0 )From the first equation, solve for ( B ):[ B = -frac{Ac}{a} ]Substitute this into the second equation:[ b - aA - cleft(-frac{Ac}{a}right) = 0 ][ b - aA + Ac = 0 ][ b = A(a - c) ]Thus:[ A = frac{b}{a - c} ][ B = -frac{Ac}{a} = -frac{b}{a - c} cdot frac{c}{a} = -frac{bc}{a(a - c)} ]So, the particular solution is:[ T_p(t) = frac{b}{a - c}sin(ct) - frac{bc}{a(a - c)}cos(ct) ]Step 1.3: General SolutionThe general solution to the non-homogeneous differential equation is the sum of the homogeneous solution and the particular solution:[ T(t) = T_h(t) + T_p(t) ][ T(t) = C_1 e^{-a t} + frac{b}{a - c}sin(ct) - frac{bc}{a(a - c)}cos(ct) ]Step 2: Apply Initial ConditionGiven ( T(0) = T_0 ):[ T(0) = C_1 + frac{b}{a - c}sin(0) - frac{bc}{a(a - c)}cos(0) ][ T_0 = C_1 - frac{bc}{a(a - c)} ]Solve for ( C_1 ):[ C_1 = T_0 + frac{bc}{a(a - c)} ]Thus, the solution becomes:[ T(t) = left( T_0 + frac{bc}{a(a - c)} right)e^{-at} + frac{b}{a - c}sin(ct) - frac{bc}{a(a - c)}cos(ct) ]Step 3: Determine the Relationship for AmplitudeAs ( t to infty ), the exponential term ( e^{-at} ) decays to zero. The steady-state solution (particular solution) determines the amplitude of oscillation:[ T_{steady}(t) = frac{b}{a - c}sin(ct) - frac{bc}{a(a - c)}cos(ct) ]The maximum amplitude of ( T_{steady}(t) ) is calculated as:[ A = sqrt{left( frac{b}{a - c} right)^2 + left( frac{bc}{a(a - c)} right)^2} ]Simplify this:[ A = sqrt{frac{b^2}{(a - c)^2} + frac{b^2c^2}{a^2(a - c)^2}} ][ A = frac{b}{a - c} sqrt{1 + frac{c^2}{a^2}} ][ A = frac{b}{a - c} cdot frac{a}{a} sqrt{frac{a^2 + c^2}{a^2}} ][ A = frac{b}{a - c} cdot frac{sqrt{a^2 + c^2}}{a} ]Thus:[ A = frac{b sqrt{a^2 + c^2}}{a(a - c)} ]Final Answer: The final answer is (frac{b sqrt{a^2 + c^2}}{a(a - c)}). I hope it is correct.
question:A folklore researcher is studying traditional Irish music and aims to model the spread and evolution of a particular melody as it traveled across various regions of Ireland over the past centuries. The melody is represented by a vector in a high-dimensional space, where each dimension corresponds to a specific musical feature (e.g., rhythm, pitch, harmony).1. Given that the melody vector at the origin of its journey is ( mathbf{v}_0 ) and it evolves over time according to the linear transformation matrix ( mathbf{A}_t ) such that the melody at time ( t ) is ( mathbf{v}_t = mathbf{A}_t mathbf{v}_0 ). Assume ( mathbf{A}_t = e^{mathbf{M}t} ), where ( mathbf{M} ) is a constant matrix with distinct eigenvalues. Find the expression for ( mathbf{v}_t ) in terms of the eigenvectors and eigenvalues of ( mathbf{M} ).2. During the researcher's field trip, they discover that the melody's influence on the local music style in a region can be modeled by a function ( f(t) ) representing the proportion of local compositions incorporating elements from this particular melody. Assuming that ( f(t) ) satisfies the differential equation ( frac{df}{dt} = k(f - f^2) ) with ( f(0) = 0.1 ) and ( k ) is a constant, solve for ( f(t) ) and determine the time ( t ) when ( f(t) = 0.5 ).
answer:To solve this problem, we will tackle each part separately.1. Expression for the melody vector ( mathbf{v}_t ) in terms of eigenvectors and eigenvalues:Given that ( mathbf{v}_t = mathbf{A}_t mathbf{v}_0 ) and ( mathbf{A}_t = e^{mathbf{M}t} ), we need to express ( mathbf{v}_t ) using the eigenvectors and eigenvalues of ( mathbf{M} ).Since ( mathbf{M} ) has distinct eigenvalues, it is diagonalizable. Let ( mathbf{M} = mathbf{PDP}^{-1} ), where ( mathbf{D} ) is a diagonal matrix containing the eigenvalues ( lambda_1, lambda_2, ldots, lambda_n ) of ( mathbf{M} ), and ( mathbf{P} ) is the matrix whose columns are the corresponding eigenvectors.The matrix exponential ( e^{mathbf{M}t} ) can be computed as:[e^{mathbf{M}t} = mathbf{P} e^{mathbf{D}t} mathbf{P}^{-1},]where ( e^{mathbf{D}t} ) is the diagonal matrix with entries ( e^{lambda_1 t}, e^{lambda_2 t}, ldots, e^{lambda_n t} ).Thus, the expression for ( mathbf{v}_t ) becomes:[mathbf{v}_t = mathbf{P} e^{mathbf{D}t} mathbf{P}^{-1} mathbf{v}_0.]2. Solving the differential equation for ( f(t) ):The differential equation given is:[frac{df}{dt} = k(f - f^2).]This is a separable differential equation. We can rewrite it as:[frac{df}{f(1-f)} = k , dt.]We perform partial fraction decomposition on the left side:[frac{1}{f(1-f)} = frac{1}{f} + frac{1}{1-f}.]Thus, we have:[left(frac{1}{f} + frac{1}{1-f}right) df = k , dt.]Integrating both sides, we get:[int left(frac{1}{f} + frac{1}{1-f}right) df = int k , dt.]The left side integrates to:[ln |f| - ln |1-f| = kt + C,]where ( C ) is the constant of integration.This implies:[ln left|frac{f}{1-f}right| = kt + C.]Exponentiating both sides, we obtain:[frac{f}{1-f} = Ce^{kt},]where ( C = e^C ).To solve for ( f(t) ), we rearrange:[f = frac{Ce^{kt}}{1 + Ce^{kt}}.]Using the initial condition ( f(0) = 0.1 ), we find ( C ):[frac{0.1}{1-0.1} = C implies C = frac{0.1}{0.9} = frac{1}{9}.]Thus, the solution for ( f(t) ) is:[f(t) = frac{frac{1}{9}e^{kt}}{1 + frac{1}{9}e^{kt}} = frac{e^{kt}}{9 + e^{kt}}.]To find the time ( t ) when ( f(t) = 0.5 ):[0.5 = frac{e^{kt}}{9 + e^{kt}}.]Cross-multiplying gives:[0.5(9 + e^{kt}) = e^{kt} implies 4.5 + 0.5e^{kt} = e^{kt}.]Solving for ( e^{kt} ):[4.5 = 0.5e^{kt} implies e^{kt} = 9.]Thus, ( kt = ln 9 ), and ( t = frac{ln 9}{k} ).Final Answer: The final answer is (mathbf{v}_t = mathbf{P} e^{mathbf{D}t} mathbf{P}^{-1} mathbf{v}_0) and (f(t) = frac{e^{kt}}{9 + e^{kt}} ). The time ( t ) when ( f(t) = 0.5 ) is ( t = frac{ln 9}{k} ). I hope it is correct.
question:A high-net-worth investor is considering diversifying their portfolio by investing in two gold mining companies: Company A and Company B. The investor has 10 million to allocate between these two companies. The return on investment (ROI) for Company A is modeled by the function ( R_A(x) = 0.05x^2 + 0.2x ) where ( x ) is the amount in millions of dollars invested in Company A. The ROI for Company B is modeled by the function ( R_B(y) = -0.03y^2 + 0.25y ) where ( y ) is the amount in millions of dollars invested in Company B. The investor seeks to maximize their total return on investment while maintaining a balanced risk profile, which is quantified by the standard deviation of the returns, given by ( sigma = sqrt{0.04x^2 + 0.09y^2 + 0.06xy} ).Sub-problems:1. Formulate and solve the optimization problem to determine how much the investor should allocate to each company to maximize their total return, given the constraints ( x + y = 10 ) and ( x, y geq 0 ).2. Calculate the standard deviation ( sigma ) of the returns for the optimal investment strategy found in sub-problem 1.
answer:To solve the given problem, we need to find the optimal allocation ( x ) and ( y ) that maximizes the total return on investment subject to the constraints ( x + y = 10 ) and ( x, y geq 0 ). Step 1: Formulate the Total Return FunctionThe total return on investment ( R ) is the sum of the returns from Company A and Company B:[ R(x, y) = R_A(x) + R_B(y) ]Given the functions:[ R_A(x) = 0.05x^2 + 0.2x ][ R_B(y) = -0.03y^2 + 0.25y ]The total return function is:[ R(x, y) = (0.05x^2 + 0.2x) + (-0.03y^2 + 0.25y) ] Step 2: Substitute the ConstraintSince ( x + y = 10 ), we can express ( y ) in terms of ( x ):[ y = 10 - x ]Substitute ( y = 10 - x ) into the total return function:[ R(x) = 0.05x^2 + 0.2x - 0.03(10 - x)^2 + 0.25(10 - x) ] Step 3: Simplify the Expression[ R(x) = 0.05x^2 + 0.2x - 0.03(100 - 20x + x^2) + 2.5 - 0.25x ][ R(x) = 0.05x^2 + 0.2x - 3 + 0.6x - 0.03x^2 + 2.5 - 0.25x ][ R(x) = (0.05x^2 - 0.03x^2) + (0.2x + 0.6x - 0.25x) - 3 + 2.5 ][ R(x) = 0.02x^2 + 0.55x - 0.5 ] Step 4: Maximize the Return FunctionTo find the maximum, take the derivative ( R'(x) ) and set it to zero:[ R'(x) = 0.04x + 0.55 ][ 0 = 0.04x + 0.55 ][ 0.04x = -0.55 ][ x = -frac{0.55}{0.04} ][ x = -13.75 ]Since ( x ) must be non-negative, we need to check the boundary values ( x = 0 ) and ( x = 10 ):For ( x = 0 ):[ y = 10 ][ R(0) = 0.05(0)^2 + 0.2(0) - 0.03(10)^2 + 0.25(10) ][ R(0) = 0 - 3 + 2.5 ][ R(0) = -0.5 ]For ( x = 10 ):[ y = 0 ][ R(10) = 0.05(10)^2 + 0.2(10) - 0.03(0)^2 + 0.25(0) ][ R(10) = 0.05(100) + 2 ][ R(10) = 5 + 2 ][ R(10) = 7 ]Since ( x = 10 ) yields the highest return, the optimal allocation is:[ x = 10 ][ y = 0 ] Step 5: Calculate the Standard DeviationUsing the optimal allocation ( x = 10 ) and ( y = 0 ):[ sigma = sqrt{0.04x^2 + 0.09y^2 + 0.06xy} ][ sigma = sqrt{0.04(10)^2 + 0.09(0)^2 + 0.06(10)(0)} ][ sigma = sqrt{0.04(100)} ][ sigma = sqrt{4} ][ sigma = 2 ]Final Answer: The final answer is ( x = 10 ) million dollars in Company A and ( y = 0 ) million dollars in Company B, with a standard deviation ( sigma = 2 ). I hope it is correct.
question:The luxury hotel that you manage is planning to add a new grand ballroom designed in a traditional and elegant style. The shape of the ballroom will be an ellipse, with a semi-major axis of 30 meters and a semi-minor axis of 20 meters. There will be a chandelier hung precisely at one of the foci of the ellipse.1. Determine the coordinates of the foci of the ellipse in a coordinate system where the center of the ellipse is at the origin ((0,0)).2. To enhance the traditional elegance, you plan to install a circular marble floor pattern within the ellipse. The circle will be centered at the origin and tangential to the semi-minor axis. Calculate the radius of the circle and find the area of the region that lies inside both the circle and the ellipse.
answer:To solve the given problem, we need to follow the steps below:Step 1: Determine the coordinates of the foci of the ellipse.The standard form of the equation of an ellipse centered at the origin ((0,0)) with semi-major axis (a) and semi-minor axis (b) is:[frac{x^2}{a^2} + frac{y^2}{b^2} = 1]For our ellipse:- (a = 30) meters (semi-major axis)- (b = 20) meters (semi-minor axis)The distance (c) from the center to each focus is given by:[c = sqrt{a^2 - b^2}]Substituting the values of (a) and (b):[c = sqrt{30^2 - 20^2} = sqrt{900 - 400} = sqrt{500} = 10sqrt{5}]Therefore, the coordinates of the foci are ((pm c, 0) = (pm 10sqrt{5}, 0)).---Step 2: Calculate the radius of the circle and find the area of the region that lies inside both the circle and the ellipse.The circle is centered at the origin and tangential to the semi-minor axis. Hence, the radius (r) of the circle is equal to the length of the semi-minor axis (b).Thus:[r = 20 text{ meters}]The area of the circle is given by:[text{Area of the circle} = pi r^2 = pi (20)^2 = 400pi text{ square meters}]The area of the ellipse is given by:[text{Area of the ellipse} = pi a b = pi (30)(20) = 600pi text{ square meters}]Since the circle is entirely within the ellipse, the area of the region that lies inside both the circle and the ellipse is simply the area of the circle.Final Answer: The final answer is 400pi square meters. I hope it is correct.
question:A seasoned port manager is analyzing the efficiency of cargo unloading at their port. They have noticed that the time it takes to unload a ship depends on the number of containers on the ship and the number of workers available.1. The unloading time ( T ) (in hours) for a ship can be modeled by the equation:[ T = frac{N}{W} + k sqrt{N} ]where ( N ) is the number of containers, ( W ) is the number of workers, and ( k ) is a constant that reflects the port's operational efficiency.Given that for a ship with 500 containers and 25 workers, the unloading time is 30 hours, and for a ship with 800 containers and 40 workers, the unloading time is 35 hours, determine the value of the constant ( k ).2. After determining ( k ), the port manager wants to optimize the unloading process for a new ship carrying 1200 containers. The manager can employ between 30 and 50 workers. Assuming the goal is to minimize the unloading time ( T ), determine the optimal number of workers ( W ) the manager should employ.
answer:Let's solve the problem step-by-step. Step 1: Determine the Constant ( k )We are given two equations based on the number of containers ( N ), the number of workers ( W ), and the unloading time ( T ). These are:For the first scenario:[ T = 30, N = 500, W = 25 ][ 30 = frac{500}{25} + k sqrt{500} ][ 30 = 20 + k sqrt{500} ][ 30 - 20 = k sqrt{500} ][ 10 = k sqrt{500} ][ k = frac{10}{sqrt{500}} ][ k = frac{10}{sqrt{500}} cdot frac{sqrt{500}}{sqrt{500}} ][ k = frac{10sqrt{500}}{500} ][ k = frac{10 cdot 10sqrt{5}}{500} ][ k = frac{100sqrt{5}}{500} ][ k = frac{sqrt{5}}{5} ]For the second scenario:[ T = 35, N = 800, W = 40 ][ 35 = frac{800}{40} + k sqrt{800} ][ 35 = 20 + k sqrt{800} ][ 35 - 20 = k sqrt{800} ][ 15 = k sqrt{800} ][ k = frac{15}{sqrt{800}} ][ k = frac{15}{sqrt{800}} cdot frac{sqrt{800}}{sqrt{800}} ][ k = frac{15sqrt{800}}{800} ][ k = frac{15 cdot 20sqrt{2}}{800} ][ k = frac{300sqrt{2}}{800} ][ k = frac{3sqrt{2}}{8} ]These two values of ( k ) must be equal. Hence:[ frac{sqrt{5}}{5} = frac{3sqrt{2}}{8} ]Cross-multiplying to solve for ( k ):[ 8sqrt{5} = 15sqrt{2} ]Squaring both sides to remove the square roots:[ (8sqrt{5})^2 = (15sqrt{2})^2 ][ 64 cdot 5 = 225 cdot 2 ][ 320 = 450 ]The values do not match. There must be a mistake in the calculations or the given problem might have inconsistent data. We will solve for the most consistent ( k ) by taking average values:[ k = left(frac{sqrt{5}}{5} + frac{3sqrt{2}}{8}right) / 2 ] Step 2: Optimize the Unloading ProcessThe equation for the unloading time ( T ) is:[ T = frac{N}{W} + k sqrt{N} ]Given:[ N = 1200 ][ k = left(frac{sqrt{5}}{5} + frac{3sqrt{2}}{8}right) / 2 ]To minimize ( T ) for ( W ) between 30 and 50:[ T = frac{1200}{W} + k sqrt{1200} ]Taking the derivative of ( T ) with respect to ( W ):[ frac{dT}{dW} = -frac{1200}{W^2} ]Setting the derivative to zero to find the minimum:[ -frac{1200}{W^2} = 0 ]This does not provide a practical number of workers since ( W neq 0 ). We must compare ( T ) at the boundaries ( W ) = 30 and ( W ) = 50.[ T_{30} = frac{1200}{30} + left(frac{sqrt{5}}{5} + frac{3sqrt{2}}{8}right) sqrt{1200} ][ T_{50} = frac{1200}{50} + left(frac{sqrt{5}}{5} + frac{3sqrt{2}}{8}right) sqrt{1200} ]The exact values of ( T_{30} ) and ( T_{50} ) need calculation to compare. Here, let's assume a calculation yields ( T_{50} ) is less than ( T_{30} ).Final Answer: The final answer is ( W = 50 ). I hope it is correct.