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?
This question is over a month old. See http://www.jiskha.com/display.cgi?id=1310699034
To find the height of the fountain, we can use the concept of trigonometry. Let's consider the diagram below:
```
|\
| \
h | \
| \
-----
R
```
In the diagram:
- h represents the height of the fountain
- R represents the radius of the circular pool
We can use the information given in the question to find the values of R and the angle θ.
1. Finding the radius of the circular pool, R:
We know that 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, the circumference of the pool is 29.0 m. So, we have:
29.0 = 2πR
To find R, we can solve the equation for R:
R = 29.0 / (2π)
2. Finding the angle θ:
The angle of elevation at the bottom of the fountain is 55.0°. Since the fountain is at the center of the pool, the angle of elevation from the edge of the pool to the top of the fountain will be half of the total angle of elevation. So, we have:
θ = 55.0° / 2
3. Finding the height of the fountain, h:
We can use the tangent function to find h. The tangent of an angle is defined as the opposite side divided by the adjacent side. Since the height h is the opposite side and the radius R is the adjacent side, we can use the formula:
tan(θ) = h / R
Substituting the known values, we have:
tan(θ) = h / (29.0 / (2π))
Simplifying the equation, we get:
h = tan(θ) * (29.0 / (2π))
Now we can calculate the height of the fountain by plugging in the values:
h = tan(55.0° / 2) * (29.0 / (2π))
Using a calculator, we can find that:
h ≈ 9.28 m
Therefore, the height of the fountain is approximately 9.28 meters.
To find the height of the fountain, we can use the trigonometric relationship between the angle of elevation and the height.
1. First, let's label the given information:
- The circumference of the pool is 29.0 m.
- The angle of elevation at the bottom of the fountain is 55.0°.
2. We need to find the radius of the circular pool. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. Rearranging the formula, we get r = C / (2π).
- Substituting the given circumference, we have r = 29.0 m / (2π).
3. The student is standing at the edge of the pool and looking at the bottom of the fountain. This forms a right-angled triangle with the height of the fountain as the opposite side, the radius of the pool as the adjacent side, and the angle of elevation as the angle between them.
4. Using trigonometric ratios, we can use the tangent function to relate the angle of elevation to the height of the fountain. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, tan(55.0°) = height / radius.
5. Rearranging the equation to solve for the height, we have height = radius * tan(55.0°).
- Substituting the calculated radius and the given angle of elevation, we can calculate the height.
6. Now, let's calculate the radius:
- r = 29.0 m / (2π) = 4.62 m (rounded to two decimal places).
7. Finally, we can find the height of the fountain:
- height = 4.62 m * tan(55.0°) = 6.79 m (rounded to two decimal places).
Therefore, the height of the fountain is approximately 6.79 meters.