# A glass vase is in the shape of a cylinder with a cone shaped opening. The height of the cylinder is 33 cm and the diameter is 20 cm. How much glass is needed to make the vase?

Jun 11, 2017

The amount of glass required is $2200 \pi c {m}^{3}$ or $6908 c {m}^{3}$.

#### Explanation:

Presuming the height of the cone also to be $33 c m$ (the presumption being that the tip of the cone-shaped opening is at the other end of the cylinder), and that the diameter is the same for both, the method we use is to subtract the volume of the cone from the volume of the cylinder - that will give us the volume of glass required.

Hence:
$V g = V c y - V c o$

The formula for volume of a cylinder is:
$V c y = \pi {r}^{2} h$, where $r =$radius, and $h =$height.

The formula for volume of a cone is:
$V c o = \pi {r}^{2} \frac{h}{3}$

Hence:
$V g = V c y - V c o$

$V g = \pi {r}^{2} h - \pi {r}^{2} \frac{h}{3}$

$V g = \pi {r}^{2} h \left(1 - \frac{1}{3}\right)$

$V g = \pi {r}^{2} h \left(\frac{2}{3}\right)$

$V g = \pi \times {10}^{2} \times 33 \times \frac{2}{3}$

Substitute $10$ for $r$ (radius is half the diameter), and $33$ for $h$.

$V g = \pi \times 100 \times 11 \cancel{33} \times \frac{2}{1 \cancel{3}}$

$V g = \pi \times 100 \times 11 \times 2$

$V g = \pi \times 2200$

$V g = 2200 \pi$

Considering $\pi = 3.14$:

$V g = 6908$