Two cylindrical jars contain the same amount of milk. If their diameters are in the ratio 3 : 4, find the ratio of their heights.

If the amount of milk is same it means volume of milk in both jars will be equal.

Let the diameters of the jars be 3a and 4a.

Radius = $\dfrac{\text{Diameter}}{2}$

Hence, radius will be $\dfrac{3a}{2}$ and $\dfrac{4a}{2}$.

Let height of the two jars be h₁ and h₂.

Volume of cylinder = πr²h

Given,

Volume of 1st jar = Volume of 2nd jar.

$\therefore π \times \Big(\dfrac{3a}{2}\Big)^2 \times h$1 = π \times \Big(\dfrac{4a}{2}\Big)^2 \times h2 \\[1em] \Rightarrow π \times \dfrac{9a^2}{4} \times h1 = π \times \dfrac{16a^2}{4} \times h2

Dividing both sides by π and multiplying by 4 we get,

$\Rightarrow 9a^2.h$1 = 16a^2.h2 \\[1em] \Rightarrow \dfrac{h1}{h2} = \dfrac{16a^2}{9a^2} \\[1em] \Rightarrow \dfrac{h1}{h2} = \dfrac{16}{9}

Hence, ratio of the heights of jars = 16 : 9.