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 18.0 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?
since c=18, r = 18/2pi
Now look at the fountain sideways. The height h can be found by noting that
h/r = tan 55.0°
Now just figure h.
To find the height of the fountain, we can use trigonometry. Let's break down the problem and explain the steps.
Step 1: Understand the problem and gather the known information.
- The student measured the circumference of the circular pool, which is 18.0 m.
- The student gauged the angle of elevation at the bottom of the fountain to be 55.0°.
Step 2: Draw a diagram to visualize the problem.
- Draw a circle representing the pool, with a dot in the center representing the fountain.
- Label the circumference of the pool as 18.0 m and the angle of elevation as 55.0°.
Step 3: Use the known 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.
- In this case, we have the value of the circumference (18.0 m), so we can rearrange the formula to solve for the radius: r = C / (2π).
Calculating the radius: r = 18.0 m / (2π) = 2.868 m (approximately)
Step 4: Use trigonometry to find the height of the fountain.
- The angle of elevation (55.0°) gives us a right triangle, with the base of the triangle being the radius of the pool (2.868 m) and the height being the height of the fountain (which we need to find).
Step 5: Identify the trigonometric relationship.
- The tangent function relates the angle of elevation (55.0°) to the ratio of opposite (height of the fountain) to adjacent (radius of the pool) sides in a right triangle.
- tan(angle) = opposite / adjacent
- In this case, tan(55.0°) = height of the fountain / 2.868 m.
Step 6: Solve the equation to find the height of the fountain.
- Rearrange the equation: height of the fountain = tan(55.0°) * 2.868 m.
Calculating the height: height of the fountain ≈ tan(55.0°) * 2.868 m ≈ 3.704 m.
Therefore, the height of the fountain is approximately 3.704 meters.