A steel storage tank for propane gas is to be constructed into the shape of a right circular cylinder with a hemisphere on each end.

Question

A steel storage tank for propane gas is to be constructed into the shape of a right circular cylinder with a hemisphere on each end. The construction cost per square foot of the end pieces is twice that of the cylindrical piece. If the desired capacity is 10pi cubic feet, what dimensions will minimize the cost?

Answer 1

The volume of the tank, if the radius of the cylinder is r and the length h, is πr²h (for the cylinder) + 4πr³/3 (for the two hemispheres, which add up to one whole sphere). We are told that this is to equal 10π, so r²h + 4r³/3 = 10 (canceling out the pi), and therefore h = 10/r² - 4r/3. Since the construction cost is double for the hemispherical ends, it is 4πr² × 2 (for the hemispheres) + 2πrh (for the cylinder), or 8πr² + 2πrh. Substituting the value of h we just found results in 8πr²+ 2πr(10/r² - 4r/3) = 16πr²/3 + 20π/r.

To minimize this, we must set its derivative with respect to r equal to zero, and then find the value of r that satisfies the resulting equation. Differentiating gives 32πr/3 - 20π/r², so 8r³/15 = 1 and r = (15/8)^1/3.Therefore, using the value for h in terms of r found previously, h = 4 × (15/8)^1/3. Therefore, the construction cost is minimized by making the length of the cylinder be four times its radius, or twice its diameter. This result applies regardless of the desired capacity of the tank.