Question

The slant height of the frustum of a cone is 4cm  and the perimeters of its circular ends are 18cm  and 6cm . Find the area of its whole surface and volume.

Answer 1

ANSWER:

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Given,

Perimeter of top circle having radius (r) is 2πr=6cm

Perimeter of bottom circle having radius (R)=18cm

Slant height of frustum of cone l=4cm

Hence,

Radius of top circle r
$= \frac{6}{2\pi} = \sqrt{ \frac{3}{\pi} } cm \\ \\ radius \: of \: bottom \: of \: \: circle \: R \: = \frac{18}{2\pi} = \frac{9}{\pi} cm$
Hight of frustum of cone is,

$h = \sqrt{ {l}^{2} - (R - r )^{2} } \\ \\ h = \sqrt{4^{2} - ( \frac{9}{\pi} \: - \frac{3}{\pi})^{2} } cm \\ \\ h = \sqrt{16 - (\frac{6}{\pi}})^{2}cm \\ \\ h = \sqrt{16 - \frac{36}{\pi ^{2} } } cm \\ \\ h = \sqrt{12.35244} cm \\ \\ h = 3.5146cm$

Now as we have the height we can calculate the curved surface area of frustum of cone.

$= r(r + r)l \\ \\ = ((r + r)l)\pi \\ \\ = (( \frac{9}{\pi} + \frac{3}{\pi} ) \times 4)\pi \: cm^{2} \\ \\ = (( \frac{12}{\pi} ) \times 4)\pi \: cm ^{2} \\ \\ =( \frac{48}{\pi})\pi {cm}^{2} \\ \\ = 48cm^{2} ........cancelling \: \pi \: \:$
Total surface area of the frustum of the cone =48cm^2.