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A circle has a radius of 9 inches. The is radius is multiplied by #2/3# to form a second circle. How is the ratio of the areas related to the ratio of the radii?

Answer 1

Avery Allen

Please refr to the Explanation.

Let, #A_i and r_i,# denote the Area and Radius of #i^(th)#

circle where, #i=1,2.#

Clearly, #A_i=pir_i^2.#

# :. A_1/A_2=(pir_1^2)/(pir_2^2)=r_1^2/r_2^2=(r_1/r_2)^2....(star).#

Given that, #r_2=2/3*r_1, or, r_1/r_2=3/2,# we have, from #(star),#

#A_1/A_2=(3/2)^2=9/4=2.25#

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Answer 2

James Atkins

The ratio of the areas of two circles is equal to the square of the ratio of their radii. Therefore, if the radius of the second circle is multiplied by ( \frac{2}{3} ) compared to the first circle, the ratio of their areas would be ( \left( \frac{2}{3} \right)^2 = \frac{4}{9} ).

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