Find the diameter of a sphere of surface area 28962 cm 2 . (round off to the closest integer)

Question

Find the diameter of a sphere of surface area 28962 cm 2 . (round off to the closest integer)

1. A. 108 cm
2. B. 48 cm
3. C. 98 cm
4. D. 96 cm (Correct answer)

Correct Answer

Option D — 96 cm

Detailed Solution & Explanation

The correct answer is 96 cm.

Shortcut Trick

• Work straight to the diameter by substituting $r = \dfrac{d}{2}$ into $S = 4\pi r^2$, which gives $S = \pi d^2$.
• $d^2 = \dfrac{28962}{\pi} = \dfrac{28962}{3.1416} \approx 9218$
• $d \approx \sqrt{9218} \approx 96.0$, so the diameter is 96 cm.

Alternate Method

• $4\pi r^2 = 28962$
• $r^2 = \dfrac{28962}{4\pi} = \dfrac{28962}{12.566} \approx 2304.8$
• $r \approx \sqrt{2304.8} \approx 48.01$ cm
• Diameter $= 2r \approx \textbf{96}$ cm

Additional Information

• $\sqrt{2304} = 48$ exactly, which is the number the question was built around — spotting that perfect square confirms the answer without a calculator.
• Sphere formulas: surface area $= 4\pi r^2$, volume $= \dfrac{4}{3}\pi r^3$. For a hemisphere, total surface area is $3\pi r^2$ (curved $2\pi r^2$ plus the flat circular face $\pi r^2$).
• Note the elegant result that a sphere's surface area equals the curved surface area of its circumscribing cylinder — both are $4\pi r^2$.