Answer to Question #151395 in Calculus for shane

Question

A cylindrical tank with a given volume V (V is constant) is to be made without a lid. The
material for the bottom costs five times as much per unit area as that for the sides. What
should be the ratio of the height to the radius for minimum cost?

Answer 1

In our case volume V is fixed V=2πr*h. In order to find the minimum price total side area is multiplied by the fixed price x and the base is multiplied by 5 times more of this price x. Total price P is: P=5xπr²+x*2πr*h.

5xπr², is the price of the base

x*2πr*h, is the price of the sides.

In order to find the minimum price we need to put V/(2πr) instead of h and take the derivative of this function with respect to variable radius and find the critical value of r by making it equal to 0.

P=5x*πr²+2x*V/r:

P'=10xπr-2x*V*r^-2==>0: solving the equation we find that critical r.

r =(V/5π)^1/3

In order to determine whether r is minimum or maximum value of radius. We need to take the second derivative of P. And if it is greater than 0 it is minimum value. If it is smaller than 0 r is maximum value. P"=10xπ+4xVr^-3

This expression is always positive since every variable is positive and they are being added. So in our case r is minimum value for the price. To find the ratio of the height to the radius. We need to find the h through r.

h=V/πr².

Ratio =(V/πr²)/r= V/πr³