Question

Jacques is an engineer. He has been given the job of designing an aluminum container having a square base and rectangular sides to hold screws and nails. It also must be open at the top. The container must use at most 750 cm2 cm 2 of aluminum, and it must hold as much as possible (i.e., have the greatest possible volume). What dimensions should the container have?

Answer 1

We let x be the length of the edge of the base of the rectangular prism and h be the height. The total surface area that needs to be covered is calculated through the equation,
A = x² + 4xh = 750
Calculating the value of h from the equation will give us,
x = (750 - x²) / 4x
The volume is equal to,
V = x²h
Substituting the expression for h, differentiating the equation and equating it to zero (concept of maxima-minima)
V = x²(750 - x²)/4x
V = 750x/4 - x³/4
dV = 750/4 - 3x²/4 = 0
x = 15.81 cm
Thus, the dimensions of the container should be 15.81 cm x 15.81 cm x 7.9 cm.