Name: Volume of Frustum of a Cone
Uploaded: 2019-07-19
Duration: 5 min 12 s
Description: In geometry, a frustum is the portion of a solid, normally a cone that lies between one or two parallel planes cutting it. Learn how to find the volume of th...

Question

The radii of the ends of a frustum of a cone $45 cm$ high are $28 cm$ and $7 cm$ . Find its volume and the curved surface area. Take ( $π = \frac{22}{7}$ )

Accepted Answer

Frustum $=$ difference of two right circular cones $O A B$ and $O C D$

Let the height of the cone $O A B$ be $h_{1}$ and its slant height $l_{1}$

i.e. $O A = O B = l_{1}$ and $O P = h_{1}$

Let $h_{2}$ be the height of cone $O C D$ and $l_{2}$ its slant height

i.e. $O C = O D = l_{2}$ and $O Q = h_{2}$

We have , $r_{1} = 28 cm$ and $r_{2} = 7 cm$

and height of frustum $(h) = 45 cm$

Also,

$h_{1} = 45 + h_{2} \dots (i)$

Now, we first need to determine the $h_{1}$ and $h_{2}$

$∵ △ O P B$ and $△ O Q D$ are similar, we have

$\frac{h_{1}}{h_{2}} = \frac{28}{7} = \frac{4}{1}$

$\Rightarrow h_{1} = 4 h_{2} \dots (ii)$

From $(i)$ and $(ii)$ , we get

$4 h_{2} = 45 + h_{2}$

$\Rightarrow 3 h_{2} = 45$

$\Rightarrow h_{2} = 15 cm$

Putting the value of $h_{2}$ in eq. $(ii)$ , we get

$h_{1} = 4 \times 15 = 60 cm$

So, $h_{1} = 60 cm$ and $h_{2} = 15 cm$

Now, the volume of frustum $=$ Vol. of cone $O A B$ $-$ Vol. of cone $O C D$

$= \frac{1}{3} π r_{1}^{2} h_{1} - \frac{1}{3} π r_{2}^{2} h_{2}$

$= \frac{1}{3} \times \frac{22}{7} [28^{2} \times 60 - 7^{2} \times 15]$

$= \frac{1}{3} \times \frac{22}{7} \times 15 [28^{2} \times 4 - 7^{2}]$

$= 5 \times \frac{22}{7} [3136 - 49]$

$= 5 \times \frac{22}{7} \times 3087$

$= 5 \times 22 \times 441$

$= 48510 {cm}^{3}$

Now, we first have to find the slant height $l_{1}$ and $l_{2}$

$l_{1} = \sqrt{{(28)^{2} + (60)^{2}}}$

$\Rightarrow l_{1} = \sqrt{(4 \times 7)^{2} + (4 \times 15)^{2}}$

$\Rightarrow l_{1} = 4 \sqrt{(7)^{2} + (15)^{2}}$

$\Rightarrow l_{1} = 4 \sqrt{49 + 225}$

$\Rightarrow l_{1} = 4 \sqrt{(274)}$

$\Rightarrow l_{1} = 4 \times 16.55$

$\Rightarrow l_{1} = 66.20 cm$

and $l_{2} = \sqrt{{(7)^{2} + (15)^{2}}}$

$\Rightarrow l_{2} = \sqrt{49 + 225}$

$\Rightarrow l_{2} = \sqrt{(274)}$

$\Rightarrow l_{2} = 16.55 cm$

So, CSA of frustum $=$ CSA of cone $O A B$ $-$ CSA of cone $O C D$

$= π r_{1} l_{1} - π r_{2} l_{2}$

$= \frac{22}{7} [28 \times 66.20 - 7 \times 16.55]$

$= 22 [4 \times 66.20 - 16.55]$

$= 22 \times 248.25$

$= 5461.5 {cm}^{2}$ .

The radii of the ends of a frustum of a cone $45 cm$ high are $28 cm$ and $7 cm$ . Find its volume and the curved surface area. Take ( $π = \frac{22}{7}$ )

Important Questions on Surface Areas and Volumes

Learn and Prepare for any exam you want !

The radii of the ends of a frustum of a cone 45 cm high are 28 cm and 7 cm. Find its volume and the curved surface area. Take (π=227)

Important Questions on Surface Areas and Volumes

Learn and Prepare for any exam you want !

The radii of the ends of a frustum of a cone $45 cm$ high are $28 cm$ and $7 cm$ . Find its volume and the curved surface area. Take ( $π = \frac{22}{7}$ )