Answer:
The ratio of the area of the sector for circle R to the area of the sector for circle Q is 25:4.
Step-by-step explanation:
Given information: Q are R different circles. The ratio of circle Q's radius to circle R's radius is 2:5.
Let the radius of Q and R are 2x and 5x respectively.
The central angle of each circle is 75°.
The area of a sector is
![A=\pi r^2(\frac{\theta}{360^{\circ}})](https://tex.z-dn.net/?f=A%3D%5Cpi%20r%5E2%28%5Cfrac%7B%5Ctheta%7D%7B360%5E%7B%5Ccirc%7D%7D%29)
where, r is the radius of circle and θ is central angle of sector.
Area of sector of circle Q.
![A_Q=\pi (2x)^2(\frac{75^{\circ}}{360^{\circ}})](https://tex.z-dn.net/?f=A_Q%3D%5Cpi%20%282x%29%5E2%28%5Cfrac%7B75%5E%7B%5Ccirc%7D%7D%7B360%5E%7B%5Ccirc%7D%7D%29)
Area of sector of circle R.
![A_R=\pi (5x)^2(\frac{75^{\circ}}{360^{\circ}})](https://tex.z-dn.net/?f=A_R%3D%5Cpi%20%285x%29%5E2%28%5Cfrac%7B75%5E%7B%5Ccirc%7D%7D%7B360%5E%7B%5Ccirc%7D%7D%29)
The ratio of area of sector for circle R to the area of the sector for circle Q is
![\frac{A_R}{A_Q}=\frac{\pi (5x)^2(\frac{75^{\circ}}{360^{\circ}})}{\pi (2x)^2(\frac{75^{\circ}}{360^{\circ}})}](https://tex.z-dn.net/?f=%5Cfrac%7BA_R%7D%7BA_Q%7D%3D%5Cfrac%7B%5Cpi%20%285x%29%5E2%28%5Cfrac%7B75%5E%7B%5Ccirc%7D%7D%7B360%5E%7B%5Ccirc%7D%7D%29%7D%7B%5Cpi%20%282x%29%5E2%28%5Cfrac%7B75%5E%7B%5Ccirc%7D%7D%7B360%5E%7B%5Ccirc%7D%7D%29%7D)
![\frac{A_R}{A_Q}=\frac{25x^2}{4x^2}](https://tex.z-dn.net/?f=%5Cfrac%7BA_R%7D%7BA_Q%7D%3D%5Cfrac%7B25x%5E2%7D%7B4x%5E2%7D)
![\frac{A_R}{A_Q}=\frac{25}{4}](https://tex.z-dn.net/?f=%5Cfrac%7BA_R%7D%7BA_Q%7D%3D%5Cfrac%7B25%7D%7B4%7D)
Therefore the ratio of the area of the sector for circle R to the area of the sector for circle Q is 25:4.