The total surface of a right circular cone of slant height 20 cm is 384π cm². Calculate:

(i) its radius in cm

(ii) its volume in cm³, in terms of π

Given, slant height, l = 20 cm and total surface area of cone = 384π cm²

(i) By formula,

Total surface area = πr(l + r)

⇒ 384π = πr(20 + r)

⇒ 384 = 20r + r²

⇒ r² + 20r - 384 = 0

⇒ r² + 32r - 12r - 384 = 0

⇒ r(r + 32) - 12(r + 32) = 0

⇒ (r + 32) = 0 or (r - 12) = 0

⇒ r = - 32 or r = 12

Since, radius cannot be negative.

∴ r = 12 cm.

Hence, radius of the cone is 12 cm.

(ii) l² = r² + h²

⇒ h² = l² - r²

⇒ h² = 20² - 12²

⇒ h² = 400 - 144

⇒ h² = 256

⇒ h = $\sqrt{256} = 16 \text{ cm.}$

Volume of cone = $\dfrac{1}{3}$ πr²h

$= \dfrac{1}{3} \times π \times 12^2 \times 16 \\[1em] = \dfrac{1}{3} \times π \times 144 \times 16 \\[1em] = \dfrac{2304}{3} π \\[1em] = 768 π \text{ cm}^3$

Hence, volume of the cone is 768 π cm³.