Angles of elevation to an airplane are measured from the top and the base of a building that is 40 m tall. The angle from the top of the building is 37°, and the angle from the base of the building is 40°. Find the altitude of the airplane. (Round your answer to two decimal places.)
draw a diagram.
seen from the plane, the top and bottom of the building subtend an angle of 3°
using the law of sines, the distance from the top of the building to the plane can be found by
40/sin3° = d/sin50°
d = 585.48 m
(h-40)/d = sin37°
h = 392.35 m
To find the altitude of the airplane, we can make use of trigonometric ratios and the concept of similar triangles.
Let's start by drawing a diagram to visualize the problem. We have a building with a height of 40 m, and we are measuring the angles of elevation from both the top and base of the building to the airplane.
```
C ← airplane
/|
/ |
40 m / | h
/ |
/ |
/θ1 |
A/_____B\
```
Let's define some points on the diagram:
- A is the base of the building
- B is the top of the building
- C is the airplane
Now, let's consider the right-angled triangle ABC. The angle at A is 90 degrees.
We are given:
- Angle BAC (angle of elevation from the base) = 40 degrees (θ1)
- Angle CBA (angle of elevation from the top) = 37 degrees (θ2)
- Height of the building AB = 40 m
We can use the tangent ratio to find the altitude h. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.
For the triangle ABC, we can write the following ratios:
tan(θ1) = h / x (1)
tan(θ2) = h / (x + 40) (2)
Where x is the distance from the base of the building to the airplane.
Solving equation (1) for x, we get:
x = h / tan(θ1)
Substituting this value of x into equation (2), we can solve for h:
tan(θ2) = h / (h / tan(θ1) + 40)
Now, let's substitute the given values into the equation and solve for h:
tan(37°) = h / (h / tan(40°) + 40)
Using a calculator:
0.7536 = h / (h / 0.8391 + 40)
Cross-multiplying and rearranging the equation:
0.7536 * (h / 0.8391 + 40) = h
0.7536h / 0.8391 + 30.144 = h
0.7536h - 0.8391h + 30.144 = 0
-0.0855h + 30.144 = 0
-0.0855h = -30.144
h = -30.144 / -0.0855
h ≈ 352.76
Rounding to two decimal places, the altitude of the airplane is approximately 352.76 meters.