Question

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.

 

Answer 1

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

Radius of the sphere (r_s) = 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 (r_c) = 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

  h²    = 9 

Further, solving for h

` h = sqrt(9)`

 h = 3 cm 

Therefore, height of the cone is 3 cm  .