Question

The areas of two circles can be expressed as the ratio of 9 to 16. What is the ratio of their radii? O A. 81 to 256 O B. 4.5 to 8 O C. 3 to 4 O D. 18 to 32

Answer 1

We are told that the ratios between areas of two circles are 9 to 16, this means the following:

$\frac{A_1}{A_2}=\frac{9}{16}$

Since the area of a circle is given by the following formula:

$A=\pi r^2$

Replacing in the proportion:

$\frac{\pi r^2_{1^{}}}{\pi r^2_2}=\frac{9}{16}$

canceling out the pis:

$\frac{r^2_1}{r^2_2}=\frac{9}{16}$

Now we use the following property of exponents:

$\frac{a^2}{b^2}=(^{}\frac{a}{b})^2$

Applying the property:

$(\frac{r_1}{r_2})^2=\frac{9}{16}$

Now we take the square root to both sides:

$\frac{r_1}{r_2}=\sqrt[]{\frac{9}{16}}$

Solving:

$\frac{r_1}{r_2}=\frac{3}{4}$

Therefore, the radius is at a proportion of 3 to 4.