A circular park feature consists of two distinct regions shaped as sectors of the same circle with radius $r=20\text{ cm}$. The first sector subtends a central angle measuring 60°, while the second sector has a central angle of $\frac{\pi }{3}$ radians. What is the ratio of the area of the first sector to the area of the second sector?

Correct Option: 1:1

To find the ratio of the area of the first sector to the area of the second sector, let us first analyze the central angle of each sector.

The radius of both sectors is given as $r=20\text{cm}$.

The central angle of the first sector, ${\theta }_1$, is given in degrees:

${\theta }_1=60°$

The central angle of the second sector, ${\theta }_2$, is given in radians:

${\theta }_2=\frac{\pi }{3}\text{radians}$

To compare the two angles directly, let us convert the angle of the first sector from degrees to radians using the conversion factor $\frac{\pi }{180°}$:

${\theta }_1=60°\times \frac{\pi }{180°}=\frac{\pi }{3}\text{radians}$

Since both sectors belong to the exact same circle with radius $r$ and subtending equal central angles (${\theta }_1={\theta }_2=\frac{\pi }{3}$), their areas must also be equal.

The formula for the area of a sector with central angle $\theta$ (in radians) is:

$\text{Area}=\frac{1}{2}{r}^2\theta$

Calculating the area of the first sector ($A_1$):

$A_1=\frac{1}{2}\times {20}^2\times \frac{\pi }{3}=\frac{200\pi }{3}{\text{cm}}^2$

Calculating the area of the second sector ($A_2$):

$A_2=\frac{1}{2}\times {20}^2\times \frac{\pi }{3}=\frac{200\pi }{3}{\text{cm}}^2$

Now, finding the ratio of the area of the first sector to the second sector:

$\text{Ratio}=\frac{A_1}{A_2}=\frac{\frac{200\pi }{3}}{\frac{200\pi }{3}}=1$

Thus, the ratio of the areas is 1:1.