Question

the radius and slant height of a cone are in the ratio 3:5 and its curved surface area is 423.9 cm^2. find the volume of the cone.(π=3.14)​

Answer 1

Given,

Curved surface area = $\mathsf{423.9 \: cm^{2}}$

Radius of cone and slant height are in ratio =$\mathsf{3:5}$

let the common ratio be $\mathsf{x}$

So the ratio becomes like,

$\mathsf{3x \: and \: 5x}$

$\large \boxed{\mathsf{Curved \: surface \: area= \pi \times r \times l}}$

where r is radius and l is slant height.

so ,on placing values,

$\mathsf{423.9 = 3.14 \times 3x\times 5x}$

$\mathsf{423.9 = 47.1 x^{2}}$

$\mathsf{\dfrac{423.9}{47.1}= x^{2}}$

$\mathsf{9 =x^{2}}$

$\mathsf{\sqrt{9} = x}$

$\large \boxed{\mathsf{x= 3}}$

So ,

Radius = $\mathsf{3 \times 3 = 9cm}$

Slant height = $\mathsf{5 \times 3 = 15cm}$

So,

Height = ?

Base (radius) = 9 cm

Hypotenuse(Slant height) = 15 cm

By Pythagoras theorem ,

$Hypotenuse^{2}= base^{2}+Height^{2}$

$\mathsf{15^{2}= 9^{2}+Height^{2}}$

$\mathsf{225 = 81 +Height^{2}}$

$\mathsf{225-81 =Height^{2}}$

$\mathsf{144 =Height^{2}}$

$\mathsf{\sqrt{144} =Height}$

$\large \boxed{\mathsf{12 cm = Height}}$

FOR VOLUME,

$\large \boxed{\mathsf{Volume= \pi \times r^{2} \times h}}$

$\mathsf{Volume= 3.14 \times 9^{2} \times 12}$

$\mathsf{Volume= 3.14 \times 81 \times 12}$

$\large \boxed{\mathsf{Volume=3052.08 cm^{2}}}$