Question

If the volume of two cones are in the ratio 1 : 4 and their diameters are in the ratio 4 : 5, then the ratio of their heights, is

Options

• 1 : 5

• 5 : 4

• 5 : 16

• 25 : 64

Answer 1

The formula of the volume of a cone with base radius ‘r’ and vertical height ‘h’ is given as

Volume = `1/3 pir^2 h`

Let the volume, base radius and the height of the two cones be `V_1 , r_1,h_1` and `V_2,r_2,h2`  respectively.

It is given that the ratio between the volumes of the two cones is 1 : 4.

Since only the ratio is given, to use them in our equation we introduce a constant ‘k’.

So, `V_1`= 1k

`V_2`= 4k

It is also given that the ratio between the base diameters of the two cones is 4 : 5.

Hence the ratio between the base radius will also be 4 : 5.

Again, since only the ratio is given, to use them in our equation we introduce another constant ‘p’.

So, `r_1` = 4p

`r_2`= 5p

Substituting these values in the formula for volume of cone we get,

`("Volume of cone_1") /("Volume of  cone_2")=((pi)(4 p)(4 p)(h_1)(3))/((3)(pi)(5p)(5 p)(h_2))`

`V_1/V_2 = (16h_1)/(25h_2)`

`(1k)/(4k) = (16h_1)/(25h_2)`

`(h_1)/(h_2) = 25/64`