Question

The ratio between the diameters of two circles is 5:7. The ratio between their areas is: A. 5:7 B. 25:49 C. 7:5 D. 49:25

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Answer


The correct option is B, which is: $$ 25:49 $$

Given that the ratio of the diameters of two circles is ( 5:7 ), let's denote the diameters as ( 5x ) and ( 7x ).

Consequently, the radii of the circles will be: $$ r_1 = \frac{5x}{2} \quad \text{and} \quad r_2 = \frac{7x}{2} $$

To determine the ratio of their areas, we use the area formula for a circle, $\pi r^2$.

Therefore, the area of the first circle is: $$ \text{Area}_1 = \pi \left(\frac{5x}{2}\right)^2 $$

And the area of the second circle is: $$ \text{Area}_2 = \pi \left(\frac{7x}{2}\right)^2 $$

Now, the ratio of the areas of the two circles is calculated as follows: $$ \frac{\text{Area}_1}{\text{Area}_2} = \frac{\pi \left(\frac{5x}{2}\right)^2}{\pi \left(\frac{7x}{2}\right)^2} $$ The $\pi$ terms cancel out: $$ \frac{\left(\frac{5x}{2}\right)^2}{\left(\frac{7x}{2}\right)^2} = \frac{\frac{25x^2}{4}}{\frac{49x^2}{4}} $$ The $\frac{x^2}{4}$ terms also cancel out: $$ \frac{25}{49} $$

Hence, the ratio of their areas is: $$ \boxed{25:49} $$


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