The angle of elevation from a point on the ground to the top of a tower is 37deg5'. The angle of elevation from a point 120 feet farther back from the
tower is 29deg5'. Find the height of the tower (to the nearest foot).
did you make a sketch?
I have , top of tower -- P , bottom of tower Q
closer point A
farther point B
In triangle PBA
angle B = 29°5'
angle BAP = 142°55'
angle BPA = 8°
AB = 120
by sine law:
PA/sin 29°5' = 120/sin 8°
PA = 419.1165
now triangle APQ is right-angled
use
sin 37°5' = PQ/PA
you finish it
I got 263.
I get 252.717 or 253
To find the height of the tower, we can use the concept of trigonometry and set up a right triangle.
Let's label the point on the ground as point A, the top of the tower as point B, and the point that is 120 feet farther back from the tower as point C.
We can determine that we have two right triangles: △ABC and △ABD, where D is a point vertically below point C.
In triangle △ABC, the angle of elevation from point A to point B is given as 37 degrees 5 minutes. This means that angle ∠ABC is 37 degrees 5 minutes.
Similarly, in triangle △ABD, the angle of elevation from point D to point B is given as 29 degrees 5 minutes. This means that angle ∠ABD is 29 degrees 5 minutes.
Now, let's consider the horizontal distance from point C to point B as x. Since point C is 120 feet farther back from the tower, then the horizontal distance from C to B is x + 120.
Using trigonometry, we can establish the following relationships:
1. In triangle △ABC:
tan(37°5') = (height of tower) / (x + 120)
2. In triangle △ABD:
tan(29°5') = (height of tower) / x
To solve for the height of the tower, we need to solve these equations simultaneously.
First, let's convert the angle measurements from minutes to decimal form:
37°5' = 37 + 5/60 = 37.0833°
29°5' = 29 + 5/60 = 29.0833°
Now plug in the values into the equations:
tan(37.0833°) = (height of tower) / (x + 120)
tan(29.0833°) = (height of tower) / x
To find x and the height of the tower, we can solve the above equations using algebraic methods or use an online calculator or trigonometric table to compute the tangent values.
For this particular problem, I suggest using an online calculator or trigonometric table to find the values of tangent and then solve the equations to determine the height of the tower. The final answer should be rounded to the nearest foot.