Question

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.)

Answer 1

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))

Answer 2

Answer:

The height is h= 28/∏r∧2, The radius = (14/∏)1/3

Explanation:

Using the formula for the volume of a cylinder

V = ∏r∧2 where r = radius and h = height

Area of cylinder is

A = 2∏r h + 2∏r∧2 where r = radius and h = height

The volume of cylinder is 28 in .3

Substitute the value into the formula, we have

∏r∧2h = 28

h = 28/∏r∧2

To calculate the Area of the cylinder, we substitute the value of h into the formula of Area of cylinder, we have

A = 2∏r (28/∏r∧2) + 2∏r∧2

= 56/r + 2∏r∧2

Therefore the Area of a cylinder is

A = 56/r + 2∏r∧2

To minimize the cost, we use the value for Area of cylinder

A = 56/r + 2∏r∧2

Differentiate with respect to r

A = d/ dr (56/r +2∏r∧2)

= -56/r ∧2 + 4∏r

Then limit to 0 to calculate the radius

-56/r ∧2 + 4∏r = 0

-56 + 4 ∏r∧3 / r ∧2 =0

-56 + 4∏r∧3 = 0

4∏r∧3 = 56

Divide both sides by 4∏

r∧3 = 56/4∏

r = (14/∏)1/3

Therefore the radius is (14/∏)1/3