Question

If the ratio of radius of base and height of a cone is 5:12 and its volume is 314 cubic metre. Find its perpendicular height and slant height
($\mathrm{\pi }=3.14$)

Answer 1

The ratio of radius of base and perpendicular height of a cone is 5 : 12.

Let the radius of base and perpendicular height of the cone be 5x and 12x, respectively.

Volume of the cone = 314 m³

$$\begin{gathered} \therefore \frac{1}{3}\mathrm{\pi }{r}^{2}h=314{\mathrm{m}}^3 \\ \Rightarrow \frac{1}{3}\times 3.14\times {\left(5x\right)}^2\times 12x=314 \\ \Rightarrow 314x^3=314 \\ \Rightarrow x^3=1 \\ \Rightarrow x=1\mathrm{m} \end{gathered}$$
∴ Perependicular height of the cone = 12x = 12 × 1 = 12 m

Radius of the cone = 5x = 5 × 1 = 5 m

Now,

(Slant height)² = (Perpendicular height)² + (Radius)²

⇒ (Slant height)² = (12)² + (5)²

⇒ (Slant height)² = 144 + 25 = 169

⇒ (Slant height)² = (13)²

⇒ Slant height = 13 m

Thus, the perpendicular height and slant height of the cone is 12 m and 13 m, respectively.