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 29.0 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?
I wish I had a figure. As I read this, the radius is 29.0/2PI
h/r=tan35
As I read your 55 is at the bottom up, so his angle down must be 35, and the tangent to that is h/radius.
solve for height h
To find the height of the fountain, we need to apply some trigonometry. Let's break down the problem step-by-step.
First, let's identify the relevant information provided in the problem:
- The circumference of the circular pool is 29.0 meters.
- The angle of elevation at the bottom of the fountain is 55.0°.
Now, let's draw a diagram to visualize the problem.
B
|\
| \
| \
| \
| \ h
| \
| \
|_______\
A r C
In the diagram, we have:
- Point A: The center of the circular pool and the base of the fountain.
- Point B: The bottom of the fountain.
- Point C: An arbitrary point on the edge of the circular pool.
- Line BC: The height of the fountain (h).
- Line AC: The radius of the circular pool (r).
- Line AB: The radius plus the height, which is the hypotenuse of the right-angled triangle.
Now, let's identify the trigonometric relationship that relates the given information. In this case, we have a right-angled triangle ABC.
The trigonometric relationship we need to use is the tangent function (tan). It relates the angle of elevation (55.0°) to the opposite side (h) and the adjacent side (r).
We have:
tan(55.0°) = h / r
To isolate h, we can rearrange the equation:
h = r * tan(55.0°)
The only information missing is the radius (r) of the circular pool. But we can find it using the circumference (C) of the pool.
We know that the formula for the circumference of a circle is given by:
C = 2 * π * r
Since we have the circumference (C) as 29.0 meters, we can rearrange the equation to solve for the radius (r):
r = C / (2 * π)
Substituting this value of r in the previous equation, we can calculate the height (h) of the fountain:
h = (C / (2 * π)) * tan(55.0°)
Now, let's calculate the height of the fountain:
First, let's calculate the radius:
r = 29.0 m / (2 * 3.14159) ≈ 4.614 m
Now, let's calculate the height:
h ≈ 4.614 m * tan(55.0°) ≈ 6.612 m
Therefore, the height of the fountain is approximately 6.612 meters.