Question

Write a function for the total cost of the cylinder in terms of its radius () and its height

Answer 1

Case 1: if the whole body of the cylinder is constructed using the same material.

Assuming the material used costs k per square unit

C(r,h) = kS(r,h)

C(r,h) = 2πrhk + 2πkr^2

Case 2 : if the base and top is made with a separate material from the one used for the body.

Assuming that the cost of materials used for the base and top is k2 per square unit.

And the body is made with material that cost k1 per square unit

C(r,h) = 2πrh(k1) + 2π(k2)r^2

Write a function for the total cost of the cylinder in terms of its radius (r) and its height (h)

Explanation:

The total cost of a cylinder is a function of the total surface area and the cost of the materials used.

The Surface area function S(r,h) of a cylinder with covered top can be written as

S(r,h) = 2πrh + 2πr^2

The Cost function C(r,h) can be written for two cases.

Case 1: if the whole body of the cylinder is constructed using the same material.

Assuming the material used costs k per square unit

C(r,h) = kS(r,h)

C(r,h) = 2πrhk + 2πkr^2

Case 2 : if the base and top is made with a separate material from the one used for the body.

Assuming that the cost of materials used for the base and top is k2 per square unit.

And the body is made with material that cost k1 per square unit

C(r,h) = 2πrh(k1) + 2π(k2)r^2

Answer 2

`r = ((V_o)/(2 \pi))^(1/3)`

`h = (V_o)/(\pi r^2)= (V_o)/(\pi (((V_o)/(2 \pi))^(1/3))^2)`

Explanation:

For this case we assume that the volume of the cylinder is given by:

`V= \pi r^2 h`

If we assume that we have a volume
`V_o` as the restriction for maximum o minimum condition we can find h in terms of r like this:

`h = (V_o)/(\pi r^2)`

Now in order to find a function in terms of the cost for the cylinder we can use the surface area like this:

`S= 2\pi r h + 2\pi r^2`

We can replace h in terms of r from the previous result:

`S(r) = 2\pi (V_o)/(\pi r^2) +2\pi r^2`

`S(r)= (2V_o)/(r) + 2\pi r^2`

And in order to maximize this function we can derivate S interms of r like this:

`(dS)/(dr)= -(2V_o)/(r^2) + 4\pi r=0`

`(V_o)/(2 \pi)= r^3`

`r = ((V_o)/(2 \pi))^(1/3)`

And with r we can find h replacing this:

`h = (V_o)/(\pi r^2)= (V_o)/(\pi (((V_o)/(2 \pi))^(1/3))^2)`