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

Answer:

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

Step-by-step explanation:

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

First derivative

$A'(r) = 2\cdot \pi \cdot r -\frac{1000}{r^{2}}$

Second derivative

$A''(r) = 2\cdot \pi +\frac{2000}{r^{3}}$

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

$2\cdot \pi \cdot r -\frac{1000}{r^{2}} = 0$

$2\cdot \pi \cdot r^{3} - 1000 = 0$

$r = \sqrt[3]{\frac{1000}{2\pi} }$

$r \approx 5.419$ (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: ($r \approx 5.419$)

$A''(5.419) = 2 + \frac{2000}{5.419^{3}}$

$A''(5.419) = 14.568$

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:

$A(5.419) = \pi\cdot (5.419)^{2} + \frac{1000}{5.419}$

$A(5.419) \approx 276.791$

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