Question

Please show step by step of working out the value of r for which is A minimum and calculate the minimum surface area of the container.
The total surface area, Acm^2, of each container is modelled by function A= πr^2+100/r.

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

Answer 1

Answer:

A = 59.63cm^2

Step-by-step explanation:

You have the following function for the surface area of the container:

$A=\pi r^2+\frac{100}{r}$                (1)

where r is the radius of the cross sectional area of the container.

In order to find the minimum surface are you first calculate the derivative of A respect to r, to find the value of r that makes the surface area a minimum.

$\frac{dA}{dr}=\frac{d}{dr}[\pi r^2+\frac{100}{r}]\\\\\frac{dA}{dr}=2\pi r-\frac{100}{r^2}$         (2)

Next, you equal the expression (2) to zero and solve for r:

$2\pi r-\frac{100}{r^2}=0\\\\2\pi r=\frac{100}{r^2}\\\\r^3=\frac{50}{\pi}\\\\r=(\frac{50}{\pi})^{1/3}$

Finally, you replace the previous result in the equation (1):

$A=\pi (\frac{50}{\pi})^{2/3}+\frac{100}{(\frac{50}{\pi})^{1/3}}}$

$A=59.63$

The minimum total surface area is 59.63cm^2