A solid hemisphere has a radius of 7 cm. Find the total surface area.
A. 400 cm²
B. 462 cm²
C. 66 cm²
D. 308 cm²
Answer (2)
The problem asks for the total surface area of a solid hemisphere with a radius of 7 cm.
The formula for the total surface area of a solid hemisphere is A = 3 π r 2 .
Substitute r = 7 cm into the formula: A = 3 π ( 7 2 ) = 147 π cm 2 .
Approximate π as 7 22 and calculate the surface area: A = 147 × 7 22 = 462 cm 2 . The total surface area is 462 cm 2 .
Explanation
Problem Analysis We are given a solid hemisphere with a radius of 7 cm. We need to find the total surface area of this hemisphere.
Formula for Surface Area The total surface area of a solid hemisphere is given by the formula:
A = 3 π r 2
where r is the radius of the hemisphere.
Substitute the Radius Substitute the given radius r = 7 cm into the formula:
A = 3 π ( 7 cm ) 2
A = 3 π ( 49 cm 2 )
A = 147 π cm 2
Calculate the Surface Area Using the approximation π ≈ 7 22 , we can calculate the surface area:
A = 147 × 7 22 cm 2
A = 21 × 22 cm 2
A = 462 cm 2
Final Answer Therefore, the total surface area of the solid hemisphere is 462 cm 2 .
Examples
Understanding surface area is crucial in many real-world applications. For instance, when calculating how much paint is needed to cover a dome-shaped structure, or when determining the amount of material required to manufacture hemispherical bowls. Knowing the surface area helps in estimating costs, optimizing material usage, and ensuring designs meet specific requirements. This concept is also vital in fields like architecture, engineering, and manufacturing, where precise measurements and calculations are essential for successful project outcomes.
The total surface area of a solid hemisphere with a radius of 7 cm is calculated using the formula A = 3πr². By substituting r = 7 cm, we find that the total surface area is approximately 462 cm². The chosen option is B: 462 cm².
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