Question

If the volumes of two cones are in the ratio 1:4 and their diameter are in the ratio 4:5, then write the ratio of their heights.

Answer 1

Hint- Here we will proceed by using the formula of volume of cone i.e. $\dfrac{1}{3}\pi {r^2}h$. Then we will use the given ratio of volume of both the cones and ratio of their diameter by equating such that we get the ratio of their heights.

Complete step-by-step answer:

Since we know the formula of volume of cone$ = \dfrac{1}{3}\pi {r^2}h$.
Here volume of cone 1 will be $\dfrac{1}{3}\pi {r_1}^2{h_1}$
And volume of cone 2 will be $\dfrac{1}{3}\pi {r_{_2}}^2{h_2}$
As we are given with the ratio of volume of two cones will be-
$\dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{\dfrac{1}{3}\pi {r_1}^2{h_1}}}{{\dfrac{1}{3}\pi {r_2}^2{h_2}}}$
Or ${\left( {\dfrac{{{r_1}}}{{{r_2}}}} \right)^2}\dfrac{{{h_1}}}{{{h_2}}}$
Now substituting the ratio of their volumes of 2 cones and their diameter,
We get-
$\dfrac{1}{4} = {\left( {\dfrac{4}{5}} \right)^2}\dfrac{{{h_1}}}{{{h_2}}}$
Or $\dfrac{{{h_1}}}{{{h_2}}} = \dfrac{{25}}{{64}}$
$\therefore $ Ratio of heights of two cones is 25:64

Note- While solving this question, we must understand that when we substitute the value in the ratio of volume of cone, radius(r) and height(h) in both the formula of volume of cone i.e. $\dfrac{1}{3}\pi {r_1}^2{h_1}$and $\dfrac{1}{3}\pi {r_2}^2{h_2}$ will be different.