Question : The diameter of a hemisphere is equal to the diagonal of a rectangle of length 4 cm and breadth 3 cm. Find the total surface area (in cm²) of the hemisphere.
Option 1: $25 \pi$
Option 2: $\frac{75 \pi}{4}$
Option 3: $\frac{50 \pi}{4}$
Option 4: $\frac{25 \pi}{4}$
Correct Answer: $\frac{75 \pi}{4}$
Solution :
Given,
The diameter of a hemisphere is equal to the diagonal of a rectangle of length 4 cm and breadth 3 cm.
We know the total surface area (TSA) of the hemisphere (radius $r$)= $3\pi r^2$
And, the length of the diagonal of the rectangle = $\sqrt{l^2+b^2}$, where $l$ and $b$ are the length and breadth of the rectangle.
So, length of diagonal = $\sqrt{4^2+3^2}$
⇒ length of diagonal = $\sqrt{16+9}$
⇒ length of diagonal = $\sqrt{25}$
⇒ length of diagonal = 5 cm
So, the Diameter of the hemisphere = 5 cm
⇒ Radius of hemisphere = $\frac52$ cm
⇒ TSA = $3\times \pi \times (\frac52)^2$
= $\frac{3\times 25\pi}{4}$ cm
2
= $\frac{75\pi}{4}$ cm
2
Hence, the correct answer is $\frac{75\pi}{4}$.
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