The entire surface of a solid cone of base radius 3 cm and height 4 cm is equal to entire surface of a solid right circular cylinder of diameter 4 cm. Find the ratio of their

(i) curved surfaces

(ii) volumes.

Radius of base of cone (r₁) = 3 cm,

Height of cone (h₁) = 4 cm.

Slant height of cone (l) = $\sqrt{r$1^2 + h1^2}

$\sqrt{3^2 + 4^2} \\[1em] = \sqrt{9 + 16} \\[1em] = \sqrt{25} \\[1em] = 5 \text{ cm}.$

Let height of cylinder be h₂ cm and radius be r₂ cm.

r₂ = $\dfrac{4}{2}$ = 2 cm.

Given, total surface area of cylinder = total surface area of cone.

⇒ 2πr(r₂ + h₂) = πr₁(l + r₁)

⇒ 2π × 2 × (2 + h₂) = π × 3 × (5 + 3)

⇒ π(8 + 4h₂) = 24π

Dividing both sides by π,

⇒ 4h₂ = 24 - 8

⇒ 4h₂ = 16

⇒ h₂ = 4 cm.

(i) Ratio between curved surface area of cone and cylinder (Ratio) = $\dfrac{πr$1l}{2πr2h}

Putting values we get,

$\dfrac{\text{Curved Surface of Cone}}{\text{Curved Surface of Cylinder}} = \dfrac{π \times 3 \times 5}{2 \times π \times 2 \times 4} \\[1em] = \dfrac{15π}{16π} \\[1em] = 15 : 16.$

Hence, the ratio between curved surface area of cone and cylinder 15 : 16.

(ii) Ratio between their volumes = $\dfrac{\text{Vol. of Cone}}{\text{Vol. of Cylinder}}$

$= \dfrac{\dfrac{1}{3}πr$1^2h1}{πr2^2h2} \\[1em] = \dfrac{r1^2h1}{3r2^2h2} \\[1em] = \dfrac{3^2 \times 4}{3 \times 2^2 \times 4} \\[1em] = \dfrac{3}{2^2} \\[1em] = \dfrac{3}{4} \\[1em] = 3 : 4.

Hence, the ratio between volumes of cones and cylinder is 3 : 4.