you are to design an opesteeln-top rectangular stainless-steel vat. It is to have a square base and a volume o

Question

of* 32ft^3, to be welded from quarter-inch plate, and weigh no more than necessary. find the dimensions.

Answer 1

I assume the given volume of 32 ft³ is the outer volume of the vat, not its internal capacity. (I am ignoring the "quarter-inch plate" in so doing.)

Let b denote the length (in feet) of a side of the square base, and let h denote the height (in feet) of the vat. The volume V of the vat is length times width times height, so

V = b² h = 32

The weight of the vat is proportional to the surface area of the vat. For an open-top vat, this area A is given by

A = b² + 4bh

(the base and the four vertical sides).

Solve the volume relation for h in terms of b. Substitute that result for h in the expression for A, then differentiate A with respect to b. Set dA/db = 0 and solve for b. You should get b = 4 and h = 2

Answer 2

Stainless Steel Vat