Question : The slant height and radius of a right circular cone are in the ratio 29 : 20. If its volume is $4838.4 \pi ~\mathrm{cm}^3$, then its radius is:
Option 1: 28 cm
Option 2: 20 cm
Option 3: 25 cm
Option 4: 24 cm
Correct Answer: 24 cm
Solution :
The ratio of slant height and radius = 29 : 20
Let slant height($l$) be $29x$ and radius($r$) be $20x$.
Height, $h = \sqrt{l^2-r^2} = \sqrt{(29x)^2-(20x)^2} = \sqrt{(841-400)x^2}= \sqrt{441x^2} = 21x$
Volume of cone = $4838.4 \pi$
⇒ $\frac{1}{3} \pi r^2 h = 4838.4 \pi$
⇒ $\frac{1}{3}× \pi ×(20x)^2 × 21x = 4838.4 \pi$
⇒ $x^3 = \frac{4838.4×3}{400×21} = \frac{172.8}{100}=1.728$
⇒ $x=\sqrt[3]{1.728}=1.2$
Radius = $20x = 20 × 1.2 = 24$ cm
Hence, the correct answer is 24 cm.
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