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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.)

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MrRoyal

Answer:

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))

jayilych4real

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 are:

  • h = 28/πr² or 28/(π((14/π)^⅔)).
  • r = (14/π)^⅓

Calculations and Parameters:

Given that:

  • r = radius,
  • h = height of the cylinder
  • Volume, V =28in³

Then, we know that

V = πr²h ----- volume of a cylinder

πr²h = 28 ---

If we make h the subject of the formula

h = 28/πr²

Area of the cylinder is;

A = 2πr² + 2πrh

Substitute 28/πr² for h to get the 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

If we 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

Then

h = 28/πr²

h = 28/(π((14/π)^⅔))

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