The curved surface area of a cone is 4070 cm² and its diameter is 70 cm. Calculate its :

(i) slant height

(ii) height and

(iii) volume

Radius, r = $\dfrac{\text{diameter}}{2} = \dfrac{70}{2} = 35 \text{ cm.}$

(i) Curved surface area = πrl

$\Rightarrow 4070 = \dfrac{22}{7} \times 35 \times \text{l} \\[1em] \Rightarrow \text{l} = \dfrac{4070 \times 7}{22 \times 35} \\[1em] \Rightarrow \text{l} = \dfrac{28490}{770} \\[1em] \Rightarrow \text{l} = 37 \text{ cm.}$

Hence, slant height of the cone is 37 cm.

(ii) l² = r² + h²

⇒ 37² = 35² + h²

⇒ h² = 1369 - 1225

⇒ h² = 144

⇒ h = $\sqrt{144}$ = 12 cm

Hence, height of the cone is 12 cm.

(iii) Volume of cone = $\dfrac{1}{3}$ πr²h

$= \dfrac{1}{3} \times \dfrac{22}{7} \times 35^2 \times 12 \\[1em] = \dfrac{22}{21} \times 1225 \times 12 \\[1em] = \dfrac{323400}{21} \\[1em] = 15400 \text{ cm}^3$

Hence, volume of the cone is 15400 cm³.