A surveyor standing 62 meters from the base of a building measures the angle to the top of the building and finds it to be 38°. The surveyor then measures the angle to the top of the radio tower on the building and finds that it is 45°. How tall is the tadio tower?
clearly he means the height of the tower part of the setup.
The height h of the tower is found by noting that
h = 62tan45° - 62tan38°
Just draw a diagram and recall your basic trig functions.
"How tall is the tadio tower?" is not clear. Do you mean its HEIGHT (how far its top is from the ground) or its VERTICAL LENGTH? Introducing the 38° angle to the top of the building implies the latter, but this could be a red herring.
To solve this problem, we can use trigonometric ratios such as tangent and sine.
Let's consider the right triangle formed by the surveyor, the top of the building, and the base of the building.
In this triangle, the angle at the surveyor is 38°, and the opposite side is the height of the building (h). The adjacent side is the distance from the surveyor to the base of the building (62 meters). Therefore, we can use the tangent ratio:
tan(38°) = h / 62
Solving for h, we have:
h = tan(38°) * 62
Using a calculator, we find:
h ≈ 0.7804 * 62 ≈ 48.32 meters
So the height of the building is approximately 48.32 meters.
Now, let's consider the right triangle formed by the top of the building, the top of the radio tower, and the base of the building.
In this triangle, the angle at the top of the building is 45°, and the opposite side is the height of the radio tower (T). The adjacent side is the height of the building we just found (48.32 meters). Therefore, we can use the tangent ratio again:
tan(45°) = T / 48.32
Solving for T, we have:
T = tan(45°) * 48.32
Again using a calculator, we find:
T ≈ 1 * 48.32 ≈ 48.32 meters
So the height of the radio tower is approximately 48.32 meters.
To find the height of the radio tower, we can set up a right triangle using the given information. Let's call the height of the building "h" (unknown) and the height of the radio tower "x" (what we want to find).
Since the surveyor is standing 62 meters from the base of the building, the distance between the surveyor and the top of the building is the hypotenuse of the triangle. We can label this distance as "y".
Now let's look at the angles. The angle between the base of the building and the top of the building is 38°, and the angle between the base of the building and the top of the radio tower is 45°.
Using trigonometric functions, we can relate the angles and sides of the triangle:
For the first triangle (building), we have:
tan(38°) = h / y
For the second triangle (building + radio tower), we have:
tan(45°) = (h + x) / y
Simplifying these equations, we have:
h = y * tan(38°)
h + x = y * tan(45°)
Now we need to eliminate the variable "y" to solve for "x". To do that, we can set up an equation by isolating "y" in the first equation and substituting it into the second equation:
y = h / tan(38°)
h + x = (h / tan(38°)) * tan(45°)
Now we can solve for "x":
x = (h / tan(38°)) * tan(45°) - h
To determine the height of the radio tower "x", we need to know the height of the building "h". Unfortunately, the height of the building is not given in the question, so we cannot calculate the exact height of the radio tower without this additional information.