Name: Applications of Direct and Inverse Variations
Uploaded: 2019-07-19
Duration: 12 min 44 s
Description: We often come across the words direct and inverse proportions. This video will take you through everything you need to know about the applications of direct...

Question

The volume of a cone varies jointly as the height and the square of its radius. The ratio of the heights of two cones is

4 : 5

and the ratio of radii is

3 : 4

, find the ratio of the volumes of two cones.

Accepted Answer

Let the volume of cone $= v$

Height $= h$

And its radius $= r$

We know that, if a variable quantity $A$ jointly varies as the two variable quantities $B and C$ then $A \propto B C or A = n B C$ , where $n$ is a constant.

Now the volume of a cone varies jointly as the height and the square of its radius.

So, $v \propto h r^{2}$

$\Rightarrow v = k h r^{2}$ ( $k =$ Constant)

Now, the ratio of the heights of two cones is $4 : 5$ .

So, the height of cone 1 and cone 2 are $4 h and 5 h$ respectively.

And the ratio of radii of cone 1 and cone 2 are $3 r and 4 r$ respectively.

So, volume of cones are $v_{1} and v_{2}$ .

$v_{1} = k (4 h) {(3 r)}^{2}$

$\Rightarrow v_{1} = k \times 4 h \times 9 r^{2}$

$\Rightarrow v_{1} = 36 k h r^{2}$ -----(i)

And $v_{2} = k (5 h) {(4 r)}^{2}$

$\Rightarrow v_{2} = k \times 5 h \times 16 r^{2}$

$\Rightarrow v_{2} = 80 k h r^{2}$ -----(ii)

So, ratio of the volumes of cones $v_{1} and v_{2}$ is $v_{1} : v_{2}$ .

$\Rightarrow 36 k h r^{2} : 80 k h r^{2}$

$\Rightarrow 9 : 20$

Hence, the ratio of the volumes of two cones is $9 : 20$ .

The volume of a cone varies jointly as the height and the square of its radius. The ratio of the heights of two cones is $4 : 5$ and the ratio of radii is $3 : 4$ , find the ratio of the volumes of two cones.

Important Questions on Variation

Learn and Prepare for any exam you want !

The volume of a cone varies jointly as the height and the square of its radius. The ratio of the heights of two cones is 4:5 and the ratio of radii is 3:4, find the ratio of the volumes of two cones.

Important Questions on Variation

Learn and Prepare for any exam you want !

The volume of a cone varies jointly as the height and the square of its radius. The ratio of the heights of two cones is $4 : 5$ and the ratio of radii is $3 : 4$ , find the ratio of the volumes of two cones.