A solid consists of a cone on top of a cylinder with a radius equal to that of the cone. The height of the cone is 42  and the height of the cylinder is 1 . If the volume of the solid is 225 pi, what is the area of the base of the cylinder?

May 3, 2016

This question is much easier than it sounds. It involves some basic algebra and simple calculations. Not all what you might expect!

Area of base = $15 \pi$

Explanation:

The cylinder and the cone have the same radius, but we do not have the length.
We have the height of the cone and the cylinder.
We know the total volume of the shape.

Write down a word equation;
vol of cylinder + vol cone = Vol of shape

Now use the formulae: $\pi {r}^{2} h + \frac{\pi {r}^{2} H}{3} = 225 \pi$

Factorise (common factor): $\pi {r}^{2} \left(h + \frac{H}{3}\right) = 225 \pi$

Divide both sides by $\pi$ and substitute: ${r}^{2} \left(1 + \frac{42}{3}\right) = 225$

Simplify: ${r}^{2} \times 15 = 225$

This gives us: ${r}^{2}$ = $\frac{225}{15}$ = 15

We could calculate the radius, but it is not necessary, we need ${r}^{2}$ to find the area of the base which is a circle.

Area of the base of thecylinder = $\pi {r}^{2} = 15 \pi$

Let's check: $\pi {r}^{2} h + \frac{\pi {r}^{2} H}{3}$

= $15 \pi \left(1\right) + \frac{15 \pi \left(42\right)}{3}$

=$15 \pi + 210 \pi$

=$225 \pi$