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Two semicircles have perimeters in a ratio of 3:8. Which of the following could be the ratios of their respective areas?

A) 1:2
B) 3:8
C) 2:1
D) 4:9

1 Answer

5 votes

Final answer:

When the perimeters of two semicircles have a ratio of 3:8, the ratio of their areas is 9:64, but this option is not available in the given choices.

Step-by-step explanation:

The question pertains to finding the ratio of the areas of two semicircles when the ratio of their perimeters is given as 3:8. Since the perimeter of a semicircle is related to its radius, it can be inferred that the radii of the semicircles are also in the ratio of 3:8. Therefore, we can determine the ratio of their areas by squaring the ratio of their radii.

The area of a circle is πr², and for a semicircle, it would be (πr²)/2. If we have two semicircles with radii r1 and r2, and their radii are in the ratio of 3:8, the ratio of their areas would be (3²):(8²) because the semicircle's area is proportional to the square of its radius. Squaring the ratio gives us 9:64, which simplifies to the ratio of areas between two semicircles having perimeters in the ratio of 3:8.

Among the provided answer choices, none accurately represents the ratio 9:64.

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