Question

Please show step by step of working out the value of r for which is A is aminimum and calculate the minimum surface area of the container.

The total surface area, of each container is modelled by function A= πr^2+1000/r.

(remember to use the derivative to show you have found the minimum)​

Answer 1

The minimum surface area of the container is 276.791 square units.

Explanation:

Let be
`A(r) = \pi\cdot r^(2) + (1000)/(r)`,
`\forall \,r \in \mathbb{R}`,
`r \geq 0`. The first and second derivatives of such function are, respectively:

First derivative

`A'(r) = 2\cdot \pi \cdot r -(1000)/(r^(2))`

Second derivative

`A''(r) = 2\cdot \pi +(2000)/(r^(3))`

The critical values of
`r` are determined by equalizing first derivative to zero and solving it: (First Derivative Test)

`2\cdot \pi \cdot r -(1000)/(r^(2)) = 0`

`2\cdot \pi \cdot r^(3) - 1000 = 0`

`r = \sqrt[3]{(1000)/(2\pi) }`

`r \approx 5.419` (since radius is a positive variable)

To determine if critical value leads to an absolute minimum, this input must be checked in the second derivative expression: (
`r \approx 5.419`)

`A''(5.419) = 2 + (2000)/(5.419^(3))`

`A''(5.419) = 14.568`

The critical value leads to an absolute minimum, since value of the second derivative is positive.

Finally, the minimum surface area of the container is:

`A(5.419) = \pi\cdot (5.419)^(2) + (1000)/(5.419)`

`A(5.419) \approx 276.791`

The minimum surface area of the container is 276.791 square units.