If the scale factor between two circles is [tex]$\frac{2 x}{5 y}$[/tex], what is the ratio of their areas?
A. [tex]$\frac{2 x}{5 y}$[/tex]
B. [tex]$\frac{2 x^2}{5 y^2}$[/tex]
C. [tex]$\frac{4 x^2}{25 y^2}$[/tex]
D. [tex]$\frac{4 x^2}{25 y^2}$[/tex]
Answer (2)
The ratio of the areas of the two circles, based on the scale factor 5 y 2 x , is 25 y 2 4 x 2 . Therefore, the correct option is C: 25 y 2 4 x 2 .
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The scale factor between the two circles is given as 5 y 2 x , which represents the ratio of their radii.
The ratio of the areas of two circles is the square of the ratio of their radii.
Square the scale factor to find the ratio of the areas: ( 5 y 2 x ) 2 = 25 y 2 4 x 2 .
The ratio of the areas of the two circles is 25 y 2 4 x 2 .
Explanation
Problem Analysis The problem states that the scale factor between two circles is 5 y 2 x . We need to find the ratio of their areas.
Calculations Let r 1 and r 2 be the radii of the two circles. The scale factor is the ratio of their radii, so we have r 2 r 1 = 5 y 2 x The area of a circle is given by the formula A = π r 2 . Let A 1 and A 2 be the areas of the two circles. Then A 1 = π r 1 2 A 2 = π r 2 2 The ratio of their areas is A 2 A 1 = π r 2 2 π r 1 2 = r 2 2 r 1 2 = ( r 2 r 1 ) 2 Since r 2 r 1 = 5 y 2 x , we have A 2 A 1 = ( 5 y 2 x ) 2 = ( 5 y ) 2 ( 2 x ) 2 = 25 y 2 4 x 2
Final Answer The ratio of the areas of the two circles is 25 y 2 4 x 2 .
Examples
Imagine you're designing two circular gardens. If the scale factor between the gardens is 5 y 2 x , then the ratio of the areas needed for planting flowers in each garden is 25 y 2 4 x 2 . This helps you determine how much seed or ground cover you'll need for each garden based on their relative sizes.
