Question

The radius and the height of a right circular cone are in the ratio 5 : 12 and its volume is 2512 cubic cm. Find the radius and slant height of the cone. (Take π = 3.14) 

Answer 1

The ratio between radius and height = 5 : 12 

Volume = 2512 cubic cm 

Let radius (r) = 5x, height (h) = 12x and slant height = l 

  

l² = r² + h² 

${\displaystyle \Rightarrow }$ l² = (5x)² + (12x)² 

${\displaystyle \Rightarrow }$ l² = 25x² + 144x²  

${\displaystyle \Rightarrow }$ l² = 169x² 

${\displaystyle \Rightarrow }$ l = 13x  

Now volume = ${\displaystyle \frac{1}{3}\pi r^{2}h}$ 

${\displaystyle \Rightarrow \frac{1}{3}\pi r^{2}h=2512}$ 

${\displaystyle \Rightarrow \frac{1}{3}\left(3.14\right){\left(5x\right)}^2\left(12x\right)=2512}$ 

${\displaystyle \Rightarrow \frac{1}{3}\left(3.14\right)\left(300x^3\right)=2512}$ 

∴ ${\displaystyle x^3=\frac{2512\times 3}{3.14\times 300}}$

= ${\displaystyle \frac{2512\times 3\times 100}{314\times 300}}$

= 8 

${\displaystyle \Rightarrow }$ x = 2 

∴ Radius = 5x = 5 × 2 = 10 

Height = 12x = 12 × 2 = 24 cm 

Slant height = 13x = 13 × 2 = 26 cm