The perimeters of the ends of a frustum of a right circular cone are $ 44 \mathrm{~cm} $ and $ 33 \mathrm{~cm} $. If the height of the frustum be $ 16 \mathrm{~cm} $, find its volume, the slant surface and the total surface.

Given:

The perimeters of the ends of a frustum of a right circular cone are \( 44 \mathrm{~cm} \) and \( 33 \mathrm{~cm} \).

The height of the frustum be \( 16 \mathrm{~cm} \).

To do:

We have to find its volume, the slant surface and the total surface.

Solution:

Perimeter of the top of frustum $= 44\ cm$

Let $r_1$ be the radius of the top.
This implies,

$2 \pi r_1=44$

$r_{1}=\frac{44 \times 7}{2 \times 22}$

$=7 \mathrm{~cm}$

Perimeter of the bottom $=33 \mathrm{~cm}$

Let $r_2$ be the radius of the bottom.
This implies,

$2 \pi r_2=33$

$r_{2}=\frac{33 \times 7}{2 \times 22}$

$=\frac{21}{4} \mathrm{~cm}$

Height of the frustum $h=16 \mathrm{~cm}$

Slant height of the frustum $l=\sqrt{h^{2}+(r_{1}-r_{2})^{2}}$

$=\sqrt{(16)^{2}+(7-\frac{21}{4})^{2}}$

$=\sqrt{256+(\frac{7}{4})^{2}}$

$=\sqrt{256+\frac{49}{16}}$

$=\sqrt{\frac{4096+49}{16}}$

$=\sqrt{\frac{4145}{16}}$

$=\frac{\sqrt{4145}}{\sqrt{16}}$

$=\frac{64.38}{4}$

$=16.095 \mathrm{~cm}$

Volume of the frustum $=\frac{\pi}{3}(r_{1}^{2}+r_{1} r_{2}+r_{2}^{2}) h$

$=\frac{22}{7 \times 3}[(7)^{2}+7 \times \frac{21}{4}+(\frac{21}{4})^{2}] \times 16$

$=\frac{22}{21}[49+\frac{147}{4}+\frac{441}{16}] \times 16$

$=\frac{22}{21}[\frac{784+588+441}{16}] \times 16$

$=\frac{22}{21} \times \frac{1813}{16} \times 16$

$=\frac{39886}{21}$

$=1899.3 \mathrm{~cm}^{3}$

$=1900 \mathrm{~cm}^{3}$

Slant surface area $=\pi(r_{1}+r_{2}) l$

$=\frac{22}{7}(7+\frac{21}{4}) \times 16.095$

$=\frac{22}{7}(\frac{28+21}{4}) \times 16.095$

$=\frac{22}{7} \times \frac{49}{4} \times 16.095$

$=619.65 \mathrm{~cm}^{2}$

Total surface area $=$ Slant surface area $+\pi r_{1}^{2}+\pi r_{2}^{2}$

$=619.65+\frac{22}{7} \times(7)^{2}+\frac{22}{7} \times(\frac{21}{4})^{2}$

$=619.65+154+\frac{22 \times 21 \times 21}{7 \times 4 \times 4}$

$=619.65+154+\frac{693}{8}$

$=619.65+154+86.625$

$=860.275 \mathrm{~cm}^{2}$

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Updated on: 10-Oct-2022

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