Question

A clown’s cap is in the shape of a cone of height 8cm and diameter 12cm. A cut is made from the apex of the cone such that slant height of the newly formed cone is 5cm. Find the ratio of volume of the frustum of the cone formed due to the cut to the volume of the cap.

• A. 0.3
• B. 0.325
• C. 0.35
• D. 0.875

Answer 1

The correct option is D

0.875

Slant height of the cap = 10cm

Let r be the smaller radius of the frustum and h be height of the frustum.

$\angle$ ABC = $\angle$ ADE = ${90}^{\circ }$
(right circular cone)
$\angle$A = $\angle$A (common angle)
Therefore $△$ABC ~ $△$ADE
(by AA similarity)

So, $\frac{AB}{AD}=\frac{BC}{DE}=\frac{AC}{AE}$ $\Rightarrow \frac{AB}{8}=\frac{r}{6}=\frac{5}{10}$

So height of the frustum AB =
$\frac{40}{10}=4cm$

Therefore , AB = 4cm and r = 3cm

Ratio of volume of frustum to the cap =
$\frac{Volumeofthefrustum}{Volumeofthecap}$

= $\frac{\frac{1}{3}\pi h({r_1}^2+{r_2}^2+r_1{r}_2)}{\frac{1}{3}\pi R^{2}H}$

= $\frac{h({r_1}^2+{r_2}^2+r_1{r}_2)}{R^{2}H}$

= $\frac{4(36+9+18)}{36\times 8}$

= $\frac{4\times 63}{36\times 8}=\frac{7}{8}=0.875$