Question : A sphere has the same curved surface area as a cone, with a vertical height of 40 cm and a radius of 30 cm. The radius of the sphere is:
Option 1: $5 \sqrt 5 \text{ cm}$
Option 2: $5\sqrt 3\text{ cm}$
Option 3: $5\sqrt{15}\text{ cm}$
Option 4: $5\sqrt{10}\text{ cm}$
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Correct Answer: $5\sqrt{15}\text{ cm}$
Solution :
Given: Curved surface area of sphere = curved surface area of cone
Height of cone = 40 cm
Radius of cone = 30 cm
The slant height $l$ of the cone = $\sqrt{h^2+r^2}$
So, $l=\sqrt{(40)^2+(30)^2}=\sqrt{1600+900}=\sqrt{2500}= 50\text{ cm}$
The curved surface area of the cone = $\pi rl$
$=\pi\times 30 \times 50=1500\pi \text{ cm}^2$
According to the question,
Curved surface area of sphere = curved surface area of cone
$4\pi r^2=1500\pi$
$⇒r=\sqrt{375}$
$⇒r=5\sqrt{15}\text{ cm}$
Hence, the correct answer is $5\sqrt{15}\text{ cm}$.
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