Find the capacity in litres of a conical vessel with height 12 cm, slant height 13 cm.

Given,

Height of the conical vessel (h) = 12 cm

Slant height of the conical vessel (l) = 13 cm

Let radius of the vessel = r cm

We know that,

⇒ l² = r² + h²

⇒ r² = l² - h²

⇒ r = $\sqrt{l^2 - h^2}$

Substituting values we get :

$r = \sqrt{(13)^2 - (12)^2} \\[1em] = \sqrt{169 - 144} \\[1em] = \sqrt{25} = 5 \text{ cm}$

By formula,

Capacity of the conical vessel (V) = $\dfrac{1}{3}πr^2h$

Substituting values we get :

$V = \dfrac{1}{3} \times \dfrac{22}{7} \times 5^2 \times 12 \\[1em] = \dfrac{1}{3} \times \dfrac{22}{7} \times 300 \\[1em] = \dfrac{6600}{21} \\[1em] = \dfrac{2200}{7} \\[1em] = \dfrac{2200}{7} \times \dfrac{1}{1000} \text{ l} \quad [∵ 1000 \text{ cm}^3 = 1 l] \\[1em] = \dfrac{22}{70} \text{ l} \\[1em] = \dfrac{11}{35} \text{ l}.$

Hence, capacity of the conical vessel = $\dfrac{11}{35}$ l.