Question

a circular fountain has a radius of 12 m and a central angle of 75°.
a) find the area of the sector.
b) find the length of the arc.

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector with radius \( r \) and central angle \( \theta \) (in degrees) is \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \). Here, \( r = 12\space m \) and \( \theta=75^{\circ} \).
Substitute the values into the formula: \( A=\frac{75^{\circ}}{360^{\circ}}\times\pi\times(12)^{2} \).
First, calculate \( (12)^{2}=144 \). Then, \( \frac{75}{360}=\frac{5}{24} \). So, \( A = \frac{5}{24}\times\pi\times144 \).
Simplify \( \frac{5}{24}\times144 = 5\times6 = 30 \). So, the area of the sector \( A = 30\pi\approx30\times3.14 = 94.2\space m^{2} \).

Step2: Recall the formula for the length of an arc

The formula for the length of an arc with radius \( r \) and central angle \( \theta \) (in degrees) is \( L=\frac{\theta}{360^{\circ}}\times2\pi r \).
Substitute \( r = 12\space m \) and \( \theta = 75^{\circ} \) into the formula: \( L=\frac{75^{\circ}}{360^{\circ}}\times2\pi\times12 \).
Simplify \( \frac{75}{360}=\frac{5}{24} \), and \( 2\times12 = 24 \). So, \( L=\frac{5}{24}\times24\pi = 5\pi\approx5\times3.14 = 15.7\space m \).

Answer:

a) The area of the sector is \( 30\pi\space m^{2}\approx94.2\space m^{2} \).
b) The length of the arc is \( 5\pi\space m\approx15.7\space m \).