Question : The radius and height of a right circular cone are in the ratio 1 : 2.4. If its curved surface area is 2502.5 cm2, then what is its volume? (Take $\pi=\frac{22}{7}$)
Option 1: 8085 cm3
Option 2: 8820 cm3
Option 3: 11550 cm3
Option 4: 13475 cm3
Correct Answer: 13475 cm 3
Solution : The ratio of the radius and height of a right circular cone = 1 : 2.4 = 10 : 24 = 5 : 12 Let radius, $r = 5x$, height, $h=12x$ and slant height be $l$. $l^2 = r^2 + h^2 = (5x)^2 + (12x)^2 = 25x + 144x = 169x^2$ ⇒ $l = \sqrt{169x^2}=13x$ Curved surface area of a cone = $\pi r l$ ⇒ $2502.5 = \frac{22}{7} × 5x × 13x$ ⇒ $x^2 = \frac{2502.5 × 7}{22 × 5 × 13} = 12.25$ ⇒ $x = 3.5$ Now, the volume of a cone = $\frac{1}{3}\pi r^2h$ = $\frac{1}{3}×\frac{22}{7}×(5x)^2×12x$ = $\frac{1}{3}×\frac{22}{7}×25× 3.5^2 ×12×3.5$ = 13475 cm 3 Hence, the correct answer is 13475 cm 3 .
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