# 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 33  and the height of the cylinder is 5 . If the volume of the solid is 226 pi, what is the area of the base of the cylinder?

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May 22, 2018

Area of the base of cylinder is $44.37$ sq.unit.

#### Explanation:

Let the radius of cylinder and cone be $r$ unit

Height of cylinder and cone are ${h}_{c} = 5 , {h}_{c n} = 33$ unit

Volume of cylinder is ${V}_{c} = \pi \cdot {r}^{2} \cdot {h}_{c} = 5 \pi {r}^{2}$

Volume of cone is ${V}_{c n} = \frac{1}{3} \pi \cdot {r}^{2} \cdot {h}_{c n} = \frac{1}{3} \cdot 33 \pi {r}^{2}$ or

${V}_{c n} = 11 \pi {r}^{2} \therefore$ Volume of composite solid is

$V = 5 \pi {r}^{2} + 11 \pi {r}^{2} = 16 \pi {r}^{2}$, which is equal to $226 \pi$

$\therefore 16 \cancel{\pi} {r}^{2} = 226 \cancel{\pi} \therefore {r}^{2} = \frac{226}{16} = \frac{113}{8}$

$\therefore r = \sqrt{\frac{113}{8}} \approx 3.7583$ . Area of the base of cylinder is

$A = \pi \cdot {r}^{2} = \pi \cdot \frac{113}{8} \approx 44.37 \left(2 \mathrm{dp}\right)$ sq.unit [Ans]

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