Question

what is the ratio of the area of sector abc to the area of sector dbe? image shows two sectors: sector abc with central angle β° and radius 2r; sector dbe with central angle 3β° and radius r. multiple-choice options: a. \\(\frac{4}{3}\\), b. \\(\frac{3}{4}\\), c. \\(\frac{2}{3}\\), d. \\(\frac{1}{3}\\), e. \\(\frac{1}{4}\\)

Explanation:

Step1: Recall the formula for the area of a sector

The area of a sector with radius \( r \) and central angle \( \theta \) (in degrees) is given by \( A=\frac{\theta}{360}\times\pi r^{2} \).

Step2: Find the area of sector \( ABC \)

For sector \( ABC \), the radius \( r_1 = 2r \) and the central angle \( \theta_1=\beta^{\circ} \). Using the sector area formula:
\( A_{ABC}=\frac{\beta}{360}\times\pi(2r)^{2}=\frac{\beta}{360}\times\pi\times 4r^{2}=\frac{4\beta\pi r^{2}}{360} \)

Step3: Find the area of sector \( DBE \)

For sector \( DBE \), the radius \( r_2 = r \) and the central angle \( \theta_2 = 3\beta^{\circ} \). Using the sector area formula:
\( A_{DBE}=\frac{3\beta}{360}\times\pi r^{2}=\frac{3\beta\pi r^{2}}{360} \)

Step4: Find the ratio \( \frac{A_{ABC}}{A_{DBE}} \)

Divide the area of sector \( ABC \) by the area of sector \( DBE \):
\( \frac{A_{ABC}}{A_{DBE}}=\frac{\frac{4\beta\pi r^{2}}{360}}{\frac{3\beta\pi r^{2}}{360}} \)
The \( \frac{\beta\pi r^{2}}{360} \) terms cancel out, leaving \( \frac{4}{3} \)? Wait, no, wait. Wait, let's re - calculate. Wait, \( A_{ABC}=\frac{\beta}{360}\times\pi(2r)^{2}=\frac{\beta}{360}\times4\pi r^{2} \), \( A_{DBE}=\frac{3\beta}{360}\times\pi r^{2} \). So the ratio is \( \frac{\frac{4\beta\pi r^{2}}{360}}{\frac{3\beta\pi r^{2}}{360}}=\frac{4\beta}{3\beta}=\frac{4}{3} \)? Wait, no, that can't be. Wait, maybe I mixed up the angles. Wait, the central angle of \( ABC \) is \( \beta \), and \( DBE \) is \( 3\beta \). Radius of \( ABC \) is \( 2r \), radius of \( DBE \) is \( r \).

Wait, let's do it again. Sector area formula: \( A=\frac{\theta}{2\pi}\times\pi r^{2}=\frac{\theta r^{2}}{2} \) (when \( \theta \) is in radians) or \( A = \frac{\theta}{360}\pi r^{2} \) (when \( \theta \) is in degrees). Let's use degrees.

\( A_{ABC}=\frac{\beta}{360}\times\pi(2r)^{2}=\frac{\beta}{360}\times4\pi r^{2} \)

\( A_{DBE}=\frac{3\beta}{360}\times\pi r^{2} \)

Now, take the ratio \( \frac{A_{ABC}}{A_{DBE}}=\frac{\frac{\beta\times4\pi r^{2}}{360}}{\frac{3\beta\times\pi r^{2}}{360}} \)

The \( \beta \), \( \pi \), \( r^{2} \), and \( 360 \) terms cancel out, so we have \( \frac{4}{3} \)? But wait, the options have \( \frac{4}{3} \) as option A. Wait, but let's check again. Wait, maybe I made a mistake in the angle. Wait, the diagram: sector \( ABC \) has central angle \( \beta \), radius \( 2r \); sector \( DBE \) has central angle \( 3\beta \), radius \( r \). So the calculation is correct. So the ratio is \( \frac{4}{3} \).

Answer:

A. \(\frac{4}{3}\)