Question

A company manufactures cylindrical metal drums with open tops with a volume of 2 cubic meters. The metal used to manufacture the drums cost $2 per square meter for the sides and $3 per square meter for the bottom. What should the dimensions be of the drums to minimize the cost of metal in their production? what is the ratio height/ radius?

Answer 1

Answer:

The cost is minimized for

$r = 0.75$  and then $h = 1.76$ and the ratio height / radius = 2.3

Step-by-step explanation:

When you solve these kind of problems you always need to ask yourself what am I minimizing or maximizing, for this case you need to minimize the cost function.

Suppose that the bottom of your drum has a radius $r$ and a height and a height $h$ then the are of the circle at the bottom would be

$A_c = \pi r^2$

And the area of the sides would be given by

$A_s = 2\pi rh$

According to what they say the cost of the sides is 2$ per square meter and the cost of the bottom is 3$ per square meter.

So the cost would be given by this function.

Cost = $4\pi rh + 3\pi r^2$

But they mention that the volume = 2 cubic meters. On the other hand

Volume = $\pi r^2 h$ = 2   if we replace on Cost we get that

$\text{Cost} = \frac{8}{r} + 3\pi r^2$

When you find the derivative of that and equal to zero you will find that the the cost is minimized for

$r = 0.75$  and then $h = 1.76$ and the ratio height / radius = 2.3