what is the ratio of the area of sector abc to the area of sector dbe?\na. $\frac{4}{3}$\nb…

Explanation:

Step1: Recall sector - area formula

The area formula of a sector of a circle with radius $R$ and central - angle $\theta$ (in degrees) is $A=\frac{\theta}{360}\times\pi R^{2}$. For sector $ABC$, the radius $R_1 = 2r$ and the central - angle $\theta_1=\beta^{\circ}$. So the area of sector $ABC$, $A_1=\frac{\beta}{360}\times\pi(2r)^{2}=\frac{\beta}{360}\times4\pi r^{2}$.

Step2: Calculate area of sector $DBE$

For sector $DBE$, the radius $R_2 = r$ and the central - angle $\theta_2 = 3\beta^{\circ}$. So the area of sector $DBE$, $A_2=\frac{3\beta}{360}\times\pi r^{2}$.

Step3: Find the ratio of the areas

The ratio of the area of sector $ABC$ to the area of sector $DBE$ is $\frac{A_1}{A_2}=\frac{\frac{\beta}{360}\times4\pi r^{2}}{\frac{3\beta}{360}\times\pi r^{2}}$. Cancel out the common factors $\frac{\beta}{360}$ and $\pi r^{2}$ in the numerator and denominator. We get $\frac{A_1}{A_2}=\frac{4}{3}$.

Answer:

A. $\frac{4}{3}$