To find the height of a tower a survivor sets up his theodolite 120 meter from the base of the tower. He finds that the angle of elevation to the top of the tower is 30° . If the instrument is 4.5 meter from the ground, what is the height of the tower?
Let the height of the tower be h meters.
From the given information, we have a right triangle formed by the survivor, the ground, and the top of the tower. The distance from the instrument to the top of the tower is the height of the tower minus the height of the instrument, which is h - 4.5 meters.
We can use trigonometry to solve for h.
In a right triangle, the tangent of an angle is equal to the opposite side divided by the adjacent side.
In this case, the angle of elevation is 30 degrees, the opposite side is h - 4.5 meters, and the adjacent side is 120 meters.
So we have the equation:
tan(30°) = (h - 4.5) / 120
Now we can solve for h.
tan(30°) = (h - 4.5) / 120
tan(30°) * 120 = h - 4.5
120 tan(30°) = h - 4.5
120 * (1/√3) = h - 4.5
120/√3 = h - 4.5
120/√3 + 4.5 = h
Using a calculator, we can find that 120/√3 ≈ 69.282
Therefore, the height of the tower is approximately 69.282 + 4.5 = 73.782 meters.
So the height of the tower is approximately 73.782 meters.