An engineer is 980 ft
from the base of a fountain at Fountain Hills,
Arizona. The angle of elevation to the top of
the column of water is 29.78. The surveyor’s
angle measuring device is at the same level
as the base of the fountain
A.280
When the top of the column of water is just half as high as in part (a), find
the angle of elevation to its top.
did you make a sketch??
This is just right-angled triangle trig
tan 29.78° = h/980
h = 980tan29.78°
=appr 561 feet
so when the height is 280.5 feet
tan Ø = 280.5/980 =.28612..
Ø = 15.97°
To find the angle of elevation to the top of the column of water when it is just half as high as in part (a), we can use trigonometry.
Let's start by defining some variables:
- Let x be the height of the column of water in part (a).
- Let α be the angle of elevation to the top of the column of water in part (a).
In part (a), we are given that the engineer is 980 ft away from the base of the fountain and the angle of elevation to the top of the column of water is 29.78 degrees. This forms a right triangle with the horizontal distance of 980 ft, the vertical height of x ft, and the angle α.
Using trigonometry, we can use the tangent function to relate the angle α and the height x:
tan(α) = x / 980
Now, in part (b), we are given that the height is just half of x.
Let's call the new height y. We can relate the angle β (the angle of elevation to the top of the column of water when it is just half as high as in part a) and the height y using the same tangent function:
tan(β) = y / 980
Since y is half of x, we can write:
y = x / 2
Substituting this into the equation, we have:
tan(β) = (x / 2) / 980
Simplifying the expression, we get:
tan(β) = x / (2 * 980)
To find β, we can take the inverse tangent (arctan) on both sides of the equation:
β = arctan(x / (2 * 980))
Given that we are not given the specific value of x in part (a), we cannot calculate the exact value of β. However, if the value of x is provided, you can substitute it into the equation to find the angle of elevation β.
To find the angle of elevation to the top of the column of water when it is half as high as in part (a), we can use the concept of similar triangles.
Let's call the height of the column of water in part (a) "h". We know that the engineer is 980 ft away from the base of the fountain and the angle of elevation to the top of the water is 29.78 degrees. Therefore, we have a right triangle where the opposite side is "h" and the adjacent side is 980 ft.
Using the trigonometric function tangent, we can find the value of "h". The tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
tan(29.78) = h/980
To find the height of the column of water when it is half as high as in part (a), we just need to calculate half of the value of "h". Let's call this height "h2".
h2 = h/2
Now, to find the new angle of elevation, let's call it "θ2", we can use the same right triangle concept but with the new height "h2".
We can use the inverse tangent function (arctan) to find the angle:
θ2 = arctan(h2/980)
Substitute the value of h2 into the equation and solve for θ2:
θ2 = arctan((h/2)/980)
Note that the units of the angle will be in degrees.