A high fountain of water is located at the center of a circular pool as in Figure P1.41. Not wishing to get his feet wet, a student walks around the pool and measures its circumference to be 34.0 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?
Tan 55=height/radius
Circumf=2PI Radius or
radius=34/2PI
height= 34/2PI * tan 55
76
To find the height of the fountain, we can use trigonometry. Let's break down the problem step by step.
Step 1: Find the radius of the circular pool
The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius of the circle. In this case, we are given that the circumference of the pool is 34.0 m.
So, we can set up the equation: 34.0 m = 2πr
Step 2: Solve for the radius
To find the radius, divide both sides of the equation by 2π:
34.0 m / (2π) = r
Step 3: Calculate the radius
Using a calculator, divide 34.0 m by 2π to get the value of r.
r ≈ 5.40 m (rounded to two decimal places)
Step 4: Use trigonometry to find the height of the fountain
In Figure P1.41, the student measures the angle of elevation at the bottom of the fountain to be 55.0°. Let's assume the height of the fountain is h.
In a right-angled triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, the height of the fountain is the opposite side and the radius of the pool is the adjacent side. So, we can set up the equation:
tan(55.0°) = h / r
Step 5: Solve for the height of the fountain
To find the height, we can rearrange the equation:
h = r * tan(55.0°)
Step 6: Calculate the height
Using a calculator, multiply the radius (5.40 m) by the tangent of 55.0° to find the height of the fountain.
h ≈ 8.55 m (rounded to two decimal places)
Therefore, the height of the fountain is approximately 8.55 meters.