The total surface area of a solid hemisphere is $ 166.32 ext{ sq cm}$, find its curved surface area?

Question

The total surface area of a solid hemisphere is $ 166.32 	ext{ sq cm}$, find its curved surface area?

Examoogle · Accepted Answer

The correct answer is Option B: 110.88 sq cm. Since the ratio of Total Surface Area (TSA) to Curved Surface Area (CSA) for a solid hemisphere is $3:2$, simply multiply the TSA by $\frac{2}{3}$ to get the CSA: $166.32 	imes \frac{2}{3} = 55.44 	imes 2 = 110.88$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Since the ratio of Total Surface Area (TSA) to Curved Surface Area (CSA) for a solid hemisphere is $3:2$, simply multiply the TSA by $\frac{2}{3}$ to get the CSA: $166.32 	imes \frac{2}{3} = 55.44 	imes 2 = 110.88$.

Formula Used: $$TSA = 3\pi r^2$$ , $$CSA = 2\pi r^2$$

Given: Total surface area of a solid hemisphere = 166.32 sq cm.

Steps: Identify Formulas: The total surface area (TSA) of a solid hemisphere includes the curved surface area and the base area, given by $$TSA = 2\pi r^2 + \pi r^2 = 3\pi r^2$$. The curved surface area (CSA) is given by $$CSA = 2\pi r^2$$.. Establish the Relationship: To find the CSA from the TSA directly, we can find the ratio: $$\frac{CSA}{TSA} = \frac{2\pi r^2}{3\pi r^2} = \frac{2}{3}$$.. Final Calculation: Substitute the given TSA value of $166.32$ sq cm into the ratio relationship: $$CSA = \frac{2}{3} 	imes 166.32$$. Dividing $166.32$ by $3$ gives $55.44$, and multiplying by $2$ gives $110.88$.

Answer: B is correct because the curved surface area is exactly two-thirds of the total surface area for a solid hemisphere, resulting in 110.88 sq cm.

Exam Tip: This ratio method ($2:3$) applies to all problems involving solid hemispheres where TSA and CSA are compared, saving time on calculating the radius ($r$) explicitly.

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