Question : The volume of a solid right circular cone is $600 \pi \;\text{cm}^3$ and the diameter of its base is 30 cm. The total surface area (in cm2) of the cone is:
Option 1: $480 \pi$
Option 2: $255 \pi$
Option 3: $472 \pi$
Option 4: $496 \pi$
Correct Answer: $480 \pi$
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 $V = 600\pi\text{ cm}^3$ and the diameter of the base is 30 cm (so the radius $r$ is 15 cm).
$⇒600\pi = \frac{1}{3}\pi (15)^2 h$
$⇒h = \frac{600 \times 3}{225} = 8 \text{ cm}$
Now, $r^2 + h^2 = l^2$
$\therefore l = \sqrt{(15)^2 + (8)^2} = 17 \text{ cm}$
The total surface area $A$ of a right circular cone is $A = \pi r (r + l)$
$⇒A = \pi \times 15 \times (15 + 17) = 480\pi \text{ cm}^2$
Hence, the correct answer is $480\pi$.
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