Question

Math question

A candle has a volume of 3217 cm3. Determine the minimum amount of plastic that is needed to cover the outside of the candle for packaging and find the dimensions of the candle that produces this surface area. [Assume the candle has a cylindrical shape ] show all your steps .

Answer 1

The candle has a radius of 8 centimeters and 16 centimeters and uses an amount of approximately 1206.372 square centimeters.

Explanation:

The volume (
`V`), in cubic centimeters, and surface area (
`A_(s)`), in square centimeters, formulas for the candle are described below:

`V = \pi\cdot r^(2)\cdot h` (1)

`A_(s) = 2\pi\cdot r^(2) + 2\pi\cdot r \cdot h` (2)

Where:

`r` - Radius, in centimeters.

`h` - Height, in centimeters.

By (1) we have an expression of the height in terms of the volume and the radius of the candle:

`h = (V)/(\pi\cdot r^(2))`

By substitution in (2) we get the following formula:

`A_(s) = 2\pi \cdot r^(2) + 2\pi\cdot r\cdot \left((V)/(\pi\cdot r^(2)) \right)`

`A_(s) = 2\pi \cdot r^(2) +(2\cdot V)/(r)`

Then, we derive the formulas for the First and Second Derivative Tests:

First Derivative Test

`4\pi\cdot r -(2\cdot V)/(r^(2)) = 0`

`4\pi\cdot r^(3) - 2\cdot V = 0`

`2\pi\cdot r^(3) = V`

`r = \sqrt[3]{(V)/(2\pi) }`

There is just one result, since volume is a positive variable.

Second Derivative Test

`A_(s)'' = 4\pi + (4\cdot V)/(r^(3))`

If
`\left(r = \sqrt[3]{(V)/(2\pi)}\right)`:

`A_(s) = 4\pi + (4\cdot V)/((V)/(2\pi) )`

`A_(s) = 12\pi` (which means that the critical value leads to a minimum)

If we know that
`V = 3217\,cm^(3)`, then the dimensions for the minimum amount of plastic are:

`r = \sqrt[3]{(V)/(2\pi) }`

`r = \sqrt[3]{(3217\,cm^(3))/(2\pi)}`

`r = 8\,cm`

`h = (V)/(\pi\cdot r^(2))`

`h = (3217\,cm^(3))/(\pi\cdot (8\,cm)^(2))`

`h = 16\,cm`

And the amount of plastic needed to cover the outside of the candle for packaging is:

`A_(s) = 2\pi\cdot r^(2) + 2\pi\cdot r \cdot h`

`A_(s) = 2\pi\cdot (8\,cm)^(2) + 2\pi\cdot (8\,cm)\cdot (16\,cm)`

`A_(s) \approx 1206.372\,cm^(2)`

The candle has a radius of 8 centimeters and 16 centimeters and uses an amount of approximately 1206.372 square centimeters.