If the circumference of a circle is increased by 50\%, by what percentage does the area of the circl

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If the circumference of a circle is increased by 50\%, by what percentage does the area of the circle increase? (Express your answer as a percentage number, e.g. write 125 for 125%)

Accepted Answer

{"content":[{"content":[{"marks":[{"type":"bold"}],"text":"Step 1:","type":"text"},{"text":" Let the original radius of the circle be ","type":"text"},{"type":"math_inline","attrs":{"latex":"r"}},{"type":"text","text":" and the new radius be "},{"attrs":{"latex":"R"},"type":"math_inline"},{"text":". The circumference ","type":"text"},{"attrs":{"latex":"C = 2\pi r"},"type":"math_inline"},{"text":" is scaled by ","type":"text"},{"attrs":{"latex":"1.5"},"type":"math_inline"},{"text":" (increased by ","type":"text"},{"attrs":{"latex":"50\%"},"type":"math_inline"},{"text":"):","type":"text"}],"type":"paragraph"},{"type":"math_block","attrs":{"latex":"2\pi R = 1.5 \times 2\pi r \implies R = 1.5r = \frac{3}{2}r"}},{"content":[{"marks":[{"type":"bold"}],"text":"Step 2:","type":"text"},{"text":" The original area of the circle is ","type":"text"},{"attrs":{"latex":"A = \pi r^2"},"type":"math_inline"},{"text":". The new area is:","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"A' = \pi R^2 = \pi (1.5r)^2 = 2.25 \pi r^2 = 2.25 A"},"type":"math_block"},{"content":[{"marks":[{"type":"bold"}],"text":"Step 3:","type":"text"},{"text":" The fractional increase in area is:","type":"text"}],"type":"paragraph"},{"type":"math_block","attrs":{"latex":"\frac{A' - A}{A} = 2.25 - 1 = 1.25"}},{"type":"paragraph","content":[{"marks":[{"type":"bold"}],"text":"Step 4:","type":"text"},{"text":" Expressed as a percentage:","type":"text"}]},{"attrs":{"latex":"1.25 \times 100\% = 125\%"},"type":"math_block"},{"content":[{"text":"So the area increases by ","type":"text"},{"attrs":{"latex":"125\%"},"type":"math_inline"},{"text":".","type":"text"}],"type":"paragraph"}],"type":"doc"}

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