A high fountain of water is located at the center of a circular pool as shown in the figure below. Not wishing to get his feet wet, a student walks around the pool and measures its circumference to be 19.0 m. Next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation at the bottom of the fountain to be 55.0°. How high is the fountain?
10.71 M.
To determine the height of the fountain, we can use trigonometry.
First, let's label the important points in the diagram. The center of the circular pool is denoted as point O, and the edge of the pool where the student is standing is point A. The bottom of the fountain is point B.
We know the circumference of the pool is 19.0 m, and we can use this information to find the radius of the pool. The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. Rearranging the formula, we have r = C / (2π).
Substituting the given value, r = 19.0 m / (2π) ≈ 3.03 m.
We also know the angle of elevation from point A to point B is 55.0°. Given the height of the fountain (denoted as h), we can define a right-angled triangle, where the side opposite angle θ (55.0°) is h, and the adjacent side is r (radius of the pool).
Using the trigonometric function tangent (tan), we have the equation tan(θ) = h / r. Rearranging the equation, we can solve for h: h = r * tan(θ).
Substituting the values, h = 3.03 m * tan(55.0°).
Using a calculator, we find that tan(55.0°) ≈ 1.428.
Therefore, h ≈ 3.03 m * 1.428 ≈ 4.34 m.
Hence, the height of the fountain is approximately 4.34 meters.