Question

part b
the fountain has four nozzles at its center. each of the nozzles on the fountain will spray a flat sheet of water that hits a sector of the circular fountain with an arc measure of 25°.
describe a strategy to find the total area of water that will be sprayed by the four nozzles when the fountain is on and the total length of the fountains sides that will get wet.

Explanation:

Step1: Find the total central angle for the four sectors

Since each nozzle sprays a sector with an arc measure of \(25^{\circ}\), and there are \(4\) nozzles. The total central angle \(\theta\) for the four - sector combination is \(\theta=4\times25^{\circ}=100^{\circ}\)

Step2: Use the sector - area formula

The formula for the area of a sector of a circle is \(A = \frac{\theta}{360^{\circ}}\times\pi r^{2}\), where \(\theta\) is the central angle of the sector and \(r\) is the radius of the circle. If we assume the radius of the circular fountain is \(r\), we can substitute \(\theta = 100^{\circ}\) into the formula \(A=\frac{100^{\circ}}{360^{\circ}}\times\pi r^{2}=\frac{5}{18}\pi r^{2}\) to find the total area of water sprayed.

Step3: Use the arc - length formula

The formula for the arc length of a sector of a circle is \(L=\frac{\theta}{360^{\circ}}\times2\pi r\). Substituting \(\theta = 100^{\circ}\) into the formula \(L=\frac{100^{\circ}}{360^{\circ}}\times2\pi r=\frac{5}{9}\pi r\) to find the total length of the fountain's sides that get wet.

Answer:

To find the total area of water sprayed, use the sector - area formula \(A=\frac{\theta}{360^{\circ}}\times\pi r^{2}\) with \(\theta = 100^{\circ}\). To find the total length of the fountain's sides that get wet, use the arc - length formula \(L=\frac{\theta}{360^{\circ}}\times2\pi r\) with \(\theta = 100^{\circ}\)