Find volume of frustum

Question Sample Titled 'Find volume of frustum'

The height and the base radius of a right circular cone are ${27}$ $\text{cm}$ and $\dfrac{{81}}{{20}}$ $\text{cm}$ respectively. The figure shows a frustum which is made by cutting off the upper part of the circular cone. The height of the frustum is ${7}$ $\text{cm}$. Find the volume of the frustum.

A
$\dfrac{{35049}}{{400}}\pi$ $\text{cm}^{{3}}$
B
$\dfrac{{45849}}{{400}}\pi$ $\text{cm}^{{3}}$
C
$\dfrac{{105147}}{{400}}\pi$ $\text{cm}^{{3}}$
D
$\dfrac{{15309}}{{400}}\pi$ $\text{cm}^{{3}}$

 Let the radius of the upper part of the frustum ,i.e. , the cut smaller cone be ${x}$ $\text{cm}$ . The height of the smaller cone $={27}-{7}$ $={20}$ $\text{cm}$

${20}$ cm${27}$ cm${x}$$\dfrac{{81}}{{20}}$ cm

 Note that the two triangles in the cross section are similar. AAA ∴  $\dfrac{{x}}{{\dfrac{{81}}{{20}}}}$ $=\dfrac{{20}}{{27}}$ corr. sides, ∼△s ${x}$ $={3}$ The volume of the frustum $={(}$the volume of the larger cone${)}-{(}$the volume of the smaller cone${)}$ $=\dfrac{{1}}{{3}}\pi{\left(\dfrac{{81}}{{20}}\right)}^{{2}}{\left({27}\right)}-\dfrac{{1}}{{3}}\pi{\left({3}^{{2}}\right)}{\left({20}\right)}$ $=\dfrac{{35049}}{{400}}\pi$ $\text{cm}^{{3}}$

 Note that the smaller cone and the larger cone are similar figures,  $\dfrac{\text{volume of smaller cone}}{\text{volume of larger cone}}$ $={\left(\dfrac{\text{length of smaller cone}}{\text{length of larger cone}}\right)}^{{3}}$ $\therefore$The volume of the frustum $=\text{volume of larger cone}\times{\left({1}-\text{volume ratio of the cone}\right)}$ $=\text{volume of larger cone}\times{\left({1}-{\left(\dfrac{\text{length of smaller cone}}{\text{length of larger cone}}\right)}^{{3}}\right)}$ $=\dfrac{{1}}{{3}}\pi{\left(\dfrac{{81}}{{20}}\right)}^{{2}}{\left({27}\right)}\times{\left({1}-{\left(\dfrac{{20}}{{27}}\right)}^{{3}}\right)}$ $=\dfrac{{35049}}{{400}}\pi$ $\text{cm}^{{3}}$

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