Question

4. Consider the design of a sheet metal circular tank closed at both ends, that is required to hold a volume of 250 m3. The cost of fabricating the tank is proportional to its surface, and is estimated at $400/m2. The tank must be housed in a shed with a sloping roof, and therefore the tank's height H is limited by H (10 - D/2), where D is the tank's diameter. (a) Formulate the minimum-cost design problem (b) Solve the optimization using the optimality conditions (c) Provide a plot of the design space (D in the abscissa, H in the coordinate) that includes contours of the objective function and also the constraints. Mark with a circle the location of the optimum. As usual, label your plot and curvesa (d) Bonus point: plot arrows at the optimum point indicating the direction of the gradient of the objective function and the active constraint. The size of the arrows need not be proportional to the magnitude of the gradients-only the directions need to be correct.

Answer 1