Minimizing Costs For its beef stew, Betty Moore Company uses aluminum containers that have the form of right circular cylinders. Find the radius and height of a container if it has a capacity of 28 in.3 and is constructed using the least amount of metal. (Round your answers to two decimal places.)

h = 28/πr² or 28/(π((14/π)^⅔))

r = (14/π)^⅓

Explanation:

Given

Let r = radius, h = height of the cylinder

Volume, V =28in³

V = πr²h volume of a cylinder

πr²h = 28 --- make h the subject of formula

h = 28/πr²

Area of the the cylinder is;

A = 2πr² + 2πrh

Substitute 28/πr² for h to get area in terms of radius

A = 2πr² + 2πr(28/πr²)

A = 2πr² + 56/r

Differentiate A with respect to r

dA/dr = 4πr - 56/r²

Set dA/dr to 0 (using the least amount of metal)

4πr - 56/r² = 0

(4πr³ - 56)/r² = 0 --- Multiply through by r²

4πr³ - 56 = 0

4πr³ = 56 --- make r the subject of formula

r³ =56/4π

r³ = 14/π

r = (14/π)^⅓

h = 28/πr²

h = 28/(π((14/π)^⅔))

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