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 = $3\times \pi \times (\frac52)^2$ = $\frac{3\times 25\pi}{4}$ cm2 = $\frac{75\pi}{4}$ cm2 Hence, the correct answer is $\frac{75\pi}{4}$.
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Question : What will be the difference between the total surface area and the curved surface area of a hemisphere having a 4 cm diameter in cm2?
Option 1: $5\pi $
Option 2: $8\pi $
Option 3: $4\pi $
Option 4: $4.4\pi $
Question : The area of a square and rectangle are equal. The length of the rectangle is greater than the length of a side of the square by 10 cm and the breadth is less than 5 cm. The perimeter (in cm) of the rectangle is:
Option 1: 50
Option 2: 40
Option 3: 80
Option 4: 100
Question : The height of a cylinder is $\frac{2}{3}$rd of its diameter. Its volume is equal to the volume of a sphere whose radius is 4 cm. What is the curved surface area (in cm2) of the cylinder?
Option 1: $\frac{112}{3} \pi$
Option 2: $32 \pi$
Option 3: $\frac{128}{3} \pi$
Option 4: $40 \pi$
Question : The total surface area of a hemisphere is 462 cm2. The diameter of this hemisphere is:
Option 1: 28 cm
Option 2: 21 cm
Option 3: 7 cm
Option 4: 14 cm
Question : If the volume of a sphere is 38808 cm³, then its surface area is:
Option 1: 5554 cm²
Option 2: 5574 cm²
Option 3: 5564 cm²
Option 4: 5544 cm²
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