Find the capacity in litres of a conical vessel with

(i) radius 7 cm, slant height 25 cm

(ii) height 12 cm, slant height 13 cm.

(i) Given, l = 25 cm and r = 7 cm.

We know that,

    l² = r² + h²
⇒ 25² = 7² + h²
⇒ h² = 25² - 7²
⇒ h² = 625 - 49
⇒ h² = 576
⇒ h = $\sqrt{576}$ = 24 cm.

Volume of cone = $\dfrac{1}{3}πr^2h$

Putting values in equation we get,

Volume of cone = $\dfrac{1}{3} \times \dfrac{22}{7} \times (7)^2 \times 24 = \dfrac{22 × 49 × 24}{3 \times 7}$ = 22 × 7 × 8 = 1232 cm³.

Since 1 litre = 1000 cm³ or, 1 cm³ = $\dfrac{1}{1000}$ litre.

∴ 1232 cm³ = $1232 \times \dfrac{1}{1000}$ litre = 1.232 litre.

Hence, the volume of cone = 1.232 litre.

(ii) Given, l = 13 cm and h = 12 cm.

We know that,

    l² = r² + h²
⇒ 13² = r² + 12²
⇒ r² = 13² - 12²
⇒ r² = 169 - 144
⇒ r² = 25
⇒ r = $\sqrt{25}$ = 5 cm.

Volume of cone = $\dfrac{1}{3}πr^2h$

Putting values in equation we get,

Volume of cone = $\dfrac{1}{3} \times \dfrac{22}{7} \times (5)^2 \times 12 = \dfrac{22 × 25 × 12}{3 \times 7} = \dfrac{6600}{21}$cm³.

Since 1 litre = 1000 cm³ or, 1 cm³ = $\dfrac{1}{1000}$ litre.

∴ $\dfrac{6600}{21}$ cm³ = $\dfrac{6600}{21} \times \dfrac{1}{1000} = \dfrac{66}{210} = \dfrac{11}{35}$ litres.

Hence, the volume of cone = $\dfrac{11}{35}$ litres.