Question

1. -/1 points a high - fountain of water is located at the center of a circular pool as shown in the figure below. a student walks around the pool and measures its circumference to be 24.6 m. next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation of the top of the fountain to be φ = 58.0°. how high is the fountain?

Explanation:

Step1: Calculate the radius of the pool

The formula for the circumference of a circle is $C = 2\pi r$. Given $C=24.6$ m, we can solve for $r$. Rearranging the formula gives $r=\frac{C}{2\pi}$. Substituting $C = 24.6$ m, we have $r=\frac{24.6}{2\pi}\approx3.915$ m.

Step2: Calculate the height of the fountain

We know that $\tan\varphi=\frac{h}{r}$, where $\varphi = 58.0^{\circ}$ is the angle of elevation and $h$ is the height of the fountain and $r$ is the radius of the pool. Rearranging for $h$ gives $h = r\tan\varphi$. Substituting $r\approx3.915$ m and $\varphi = 58.0^{\circ}$, we get $h=3.915\times\tan(58.0^{\circ})\approx6.26$ m.

Answer:

$6.26$ m