A cylindrical tank is to be fabricated from a solid material under these constraints: it has a fixed inner volume of V mm³, the cylindrical wall is 2 mm thick, and the top is open. The base is a solid circular disc of thickness 2 mm whose radius equals the outer radius of the tank. If the

Question

A cylindrical tank is to be fabricated from a solid material under these constraints: it has a fixed inner volume of V mm³, the cylindrical wall is 2 mm thick, and the top is open. The base is a solid circular disc of thickness 2 mm whose radius equals the outer radius of the tank. If the volume of material used is minimized when the inner radius of the tank is 10 mm, find the value of V / (250 * pi).

Accepted Answer

4. With inner radius r and height h, inner volume V = pi * r² * h so h = V / (pi * r²). The wall occupies the annular region from r to r+2 over height h, and the base is a solid disc of radius r+2 and thickness 2. Material volume M = pi * [(r+2)² - r²] * h + pi * (r+2)² * 2. Substituting h and setting dM/dr = 0 at r = 10 gives V = 4 * 250 * pi, so V/(250*pi) = 4.

Solution

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