# The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. Find the height of the cone. - Mathematics

Short Note

The surface area of a sphere of radius 5 cm is five times the area of the curved surface of a cone of radius 4 cm. Find the height of the cone.

#### Solution

In the given problem, we are given a sphere and a cone of the following dimensions:

Radius of the sphere (rs) = 5 cm

So, surface area of the sphere = 4 pi r^2 ,

= 4 pi (5)^2

= 100 π cm2

Also, radius of the cone base (rc) = 4 cm

So, curved surface area of the cone = pi r_cl

 = 4 πl

Now, it is given that the surface area of the sphere is 5 times the curved surface are of the cone. So, we get

100 pi = (5) (4pi l)

 l=100/20

   l = 5  cm

Now, slant height (l) of a cone is given by the formula:

l = sqrt(r^2 + h^2 )

So, let us take the height of the cone as h,

We get,

5=sqrt(4)^2 +(h)^2

Squaring both sides,

(5)^2 = (sqrt(16+(h)^2))^2

25  = 16 + h2

h2   = 25-16

h   = 9

Further, solving for h

 h = sqrt(9)

h = 3 cm

Therefore, height of the cone is 3 cm  .

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#### APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 21 Surface Areas and Volume of a Sphere
Q 9 | Page 25