Question : The volume of a right circular cone is 308 cm3 and the radius of its base is 7 cm. What is the curved surface area (in cm2) of the cone? (Take $\pi=\frac{22}{7}$)
Option 1: $22 \sqrt{21}$
Option 2: $44 \sqrt{21}$
Option 3: $22 \sqrt{85}$
Option 4: $11 \sqrt{85}$
Correct Answer: $22 \sqrt{85}$
Solution :
The volume $V$ of a right circular cone is $V = \frac{1}{3}\pi r^2 h$, where $r$ is the radius of the base and $h$ is the height of the cone.
Given that $V = 308 \, \text{cm}^3$ and $r = 7 \, \text{cm}$,
$⇒308 = \frac{1}{3}\pi (7)^2 h$
$⇒h = \frac{308 \times 3}{22 \times 7} = 6 \, \text{cm}$
Slant height $l = \sqrt{r^2 + h^2}$
Substituting $r = 7 \, \text{cm}$ and $h = 6 \, \text{cm}$, we get:
$⇒l = \sqrt{(7)^2 + (6)^2} = \sqrt{85} \, \text{cm}$
Substituting $r = 7 \, \text{cm}$ and $l = \sqrt{85} \, \text{cm}$ into the formula for curved surface area of cone $ \pi rl = \pi \times 7 \times \sqrt{85} = 22\sqrt{85} \, \text{cm}^2$
Hence, the correct answer is $ 22\sqrt{85}$.
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