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 13.0 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?
The pool radius is R = 13/(2*pi) meters
The fountain height is R*tan 55
12.29m
answer is 3.0m
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ទិញដុល្លារនៅខេត្តព្រះសីហនុ
ទិញដុល្លារនៅខេត្តព្រះសីហនុ
To determine the height of the fountain, we first need to find the radius of the circular pool. We can use the circumference of the pool to do this.
The formula for the circumference of a circle is given by:
C = 2πr
where C is the circumference and r is the radius of the circle.
In this case, we are given the circumference (C) as 13.0 m. So we can rewrite the formula as:
13.0 = 2πr
Now, let's solve for r:
Divide both sides of the equation by 2π:
13.0 / (2π) = r
Using a calculator, we find:
r ≈ 2.07 m (rounded to two decimal places)
Now that we have the radius of the circular pool, we can move on to calculating the height of the fountain.
Let's consider Figure P1.41. The angle of elevation at the bottom of the fountain is given as 55.0°. The height of the fountain can be determined using trigonometry.
The trigonometric function that relates the opposite side (height of the fountain) to the adjacent side (radius of the circular pool) is the tangent function:
tan(θ) = opposite (height) / adjacent (radius)
In this case, the angle (θ) is 55.0° and the radius (adjacent side) is 2.07 m.
Now, we can rearrange the formula to solve for the height (opposite side):
height = tan(θ) * radius
Substituting the given values:
height = tan(55.0°) * 2.07
Using a calculator, we find:
height ≈ 3.30 m (rounded to two decimal places)
Therefore, the height of the fountain is approximately 3.30 m.