A high fountain of water is located at the
center of a circular pool as in the figure. A
student walks around the pool and estimates
its circumference to be 177 m. Next, the
student stands at the edge of the pool and uses
a protractor to gauge the angle of elevation of
the top of the fountain to be 45.6
◦
.
How high is the fountain?
it is on my physics assignment so, therefore its physics.
2 pi r = 177
so
r = 177/(2 pi)
then
tan 45.6 = h/r
so
h = r tan 45.6
Wondering how this is physics...
To find the height of the fountain, we can use trigonometry. We need to use the angle of elevation and the distance around the pool (circumference) to calculate the height.
Here's how we can do it step by step:
Step 1: Convert the angle of elevation from degrees to radians.
- Remember that there are 2π radians in a complete circle and 360 degrees in a circle.
- Convert 45.6 degrees to radians by multiplying it by π/180.
45.6 degrees * (π/180) = 0.79448 radians (approx)
Step 2: Calculate 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.
- Rearrange the formula to solve for the radius: r = C / (2π)
Given that the estimated circumference is 177 m:
r = 177 m / (2π) = 28.17 m (approx)
Step 3: Use trigonometry to find the height of the fountain.
- We'll use the tangent function, which relates the angle of elevation to the height and the radius of the pool.
- The tangent function is given by the formula: tan(angle) = height / radius
- Rearrange the formula to solve for the height: height = radius * tan(angle)
Substituting the values we calculated:
height = 28.17 m * tan(0.79448 radians)
Using a calculator, we can find the approximate value of the tangent:
height ≈ 28.17 m * 0.975 ≈ 27.44 m
So, the height of the fountain is approximately 27.44 meters.