Math Is Fun Forum

This is an engineering economics type problem. You are given the total volume of the tank:
(1) 200 = pi r r h + pi 4 r r r/3
where r is the unknown radius and h is the unknown height of the cylinder.

You are given the inequality:
(2) 16 > 2 r + h

You are given that Xc (the cost per unit area of the cylinder) is half of Xs (the cost per unit area of the hemispheres). Hence, the total cost is:
(3) (Ac) Xc + (As) Xs = (2 pi r h) Xs/2 + (4 pi r r) Xs = C

At this point, we want to replace the height h in equation (3). Rearranging equation (1), we get:
(1)' h = 200/(pi r r) - 4 r/3  Replacing h in equation (3) and simplifying, we get:
(4) C = Xs pi [8 r r/3 + 200/(pi r)]

Equation (4) is the cost equation in terms of the unknown radius r. This is a smooth curve with r as the independent variable, so we can apply the calculus. In order to find the minimum cost, we take the derivative of equation (4) with respect to r and set the resulting expression to zero. Thus, we get:
(5)  dC/dr = Xs pi [16 r/3 - 200/(pi r r)] = 0

Solving equation (5) for r yields the value:
r r r = 200 3/(pi 16) and the cube root yields the desired value r = 2.285 m or 228.5 cm.
Substituting this value for r into equation (1)' yields the value of h = 9.142 m or 914.2 cm.

You may verify that this solution satisfies the inequality (2) and equation (1). To convince yourself that this value of r yields the minimum cost of manufacturing the tank, substitute r = 2.285 into the expression within the square brackets of equation (4). Then repeat using other values of r in the expression, say r = 2 and r = 3.

cheers