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?

Question

Giuliana Ritter

Mathematics

27 April, 21:47

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?

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Carson Waters · 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.