A simple way to model the construction of an oil tanker is to start with a large rectangular sheet of steel that is x feet wide and 3x feet long. Now cut a smaller square that is t feet on a side out of each corner of the larger sheet and fold up and weld the sides of the steel sheet to make a traylike structure with no top.a. Show that the volume of oil that can be held by this tray is given byV = t (x -2t )(3x -2t) =3tx2-8t2x 9+4t3.b. How should tbe chosen so as to maximize V for any given value of x?c. Is there a value of x that maximizes the volume of oil that can be carried?d. Suppose that a shipbuilder is constrained to use only 1,000,000 square feet of steel sheet to construct an oil tanker. This constraint can be represented by the equation 3×2 – 4t2= 1,000,000 (because the builder can return the cut-out squares for credit). How does the solution to this constrained maximum problem compare to the solutions described in parts (b) and (c)?

Given that the length is 3x feet and the breadth is x feet. On cutting a smaller square that is t feet on the side out of each corner of a larger sheet gives a tray like structure. Thus the volume of the formed structure= length*width*height= (3x-2t)(x-2t)t= 3tx^2 -8t^2x+4t^3. The value of t such that V can be maximized can be found by dV/dt = 3x^2 16 xt +12t^2= 0. On applying the quadratic solving procedure, we have that t1= 16x + sqrt(256x^2-144x^2) / 24 = 16x + 10.6x / 24 = 0.225x t2= 16x – sqrt(256x^2-144x^2) / 24 = 16x- 10.6x/24 = 1.11x The maxima will occur where the second derivate of V wrt t has a negative sign. The second derivative of v wrt t is -16x+24t. Value of -16x+24t =…

0.225x) = -16x+5.4x=-10.6x Thus, v attains a maxima at t=0.225x. c) At t=0.225x, V= 0.67x^3 -0.04x^3 + 0.05x^3 = 0.68 x^3. Thus, v is a continuously increasing function. d) The constrained maximization exercise requires the Lagrangian method of solving the three simultaneous equations including the constraint equation. Thus the solution in the constrained optimization problem would be different from that of the unconstrained optimization held in the previous questions.