The angle of elevation from point A to the top of a vertical pole is 30.0 degrees. At point B, 6 meters closer to the base of the pole and on a line through the base of the pole and point A, the angle of elevation to the top of the pole is 37.0 degrees. Find the height ( to the nearest tenth of a foot) of the pole.
Let X be the distance from the closer point B) to the pole.
Let H be the height of the pole. You have two equations in two unknowns.
H/X = tan 37 = 0.75355
H/(X+6) = tan 30 = 0.57735
(X+6)/X = 1.3052 = 1 + 6/X
6/X = 0.3052
X = 19.66 m
H = 0.75355X = 14.8 m
To find the height of the pole, we can use simple trigonometry. Let's break down the problem and identify the given information:
1. From point A, the angle of elevation to the top of the pole is 30.0 degrees.
2. Point B is 6 meters closer to the base of the pole, and the angle of elevation from point B to the top of the pole is 37.0 degrees.
Now, let's set up a diagram to visualize the situation:
```
A B X
|\ | |
| \ | |
| \ | |
| \ | |
| \ | | h (height of the pole)
| \ | |
| \ | |
| \ | |
| \ | |
| \ | |
-----------------------|---------------
d=6m
```
In the diagram above, consider:
- Point A as the initial position of observation.
- Point B as the second position of observation, 6 meters closer to the base of the pole than point A.
- X as the top of the pole.
- h as the height of the pole.
- d as the distance between points B and X (which is the same as the distance between points A and X).
Now, let's use trigonometry to find the height of the pole. We will focus on the right triangle formed by points B, X, and the base of the pole. The tangent function will be helpful in this case.
From the right triangle BXC, we can calculate the height of the pole (h) using the tangent of the angle of elevation from point B (37.0 degrees):
tan(37.0 degrees) = h / d
However, we need to find the value of d. To do that, we can use the right triangle ABX. The tangent of the angle of elevation from point A (30.0 degrees) will help us find d:
tan(30.0 degrees) = h / (d + 6)
Now we have a system of equations. We can solve it to find h:
tan(37.0 degrees) = h / d (Equation 1)
tan(30.0 degrees) = h / (d + 6) (Equation 2)
By rearranging Equation 1, we get:
h = d * tan(37.0 degrees)
Substituting this value of h into Equation 2, we get:
tan(30.0 degrees) = (d * tan(37.0 degrees)) / (d + 6)
From this equation, we can solve for d. After finding d, we can substitute it back into Equation 1 to find the height of the pole (h).
Now, you can use a calculator or software to solve this equation and find the height of the pole to the nearest tenth of a foot.