Determine objective and constraint equations. Express quantity to be minimized as a function of x. Optimal of?

You have an open rectangukar box with a square base. Consider the problem of finding the values of x and h for which the volume is 32 cubic feet and the total surface are of the box is minimal. (The surface area is the sum of the five faces of the box.)





A. Determine the objective and constraint equations.


B. Express the quantity to be minimized as a function of x.


C. Find the optimal values of x and h.

Determine objective and constraint equations. Express quantity to be minimized as a function of x. Optimal of?
I'm going to label length "L" instead of "x"





Constraint:


V = 32 = L * L * h (L=w because the base is square)





Objective:


SA = L*L + 4*L*h (only one L*L because the box is open)





so 32/L^2 = h


therefore SA = L^2 + 4*L*(32/L^2)


= L^2 + 128/L





Take the derivative and find the critical points in order to optimize:


SA' = 2L - 128/L^2 = 0


2L(1 - 64/L^3) = 0


1 = 64/ L^3


L = 4





V = 32 = L * L * h


so 16h = 32


h = 2





Hope this helps you understand optimization a little better.