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 31°, and the angle from the base of the building is 40°. Find the altitude of the airplane. (Round your answer to two decimal places.)
measured from the plane, the building subtends an angle of 9°
the distance d from the top of the building to the plane is found by
d/sin50° = 40/sin9°
d = 195.87
now, knowing d, the height h of the plane can be found:
(h-40)/d = sin31°
(h-40)/195.87 = sin31°
h = 40 + 195.87 sin31°
h = 140.88
h =
To find the altitude of the airplane, we can use the concept of trigonometry, specifically the tangent function.
Let's denote the altitude of the airplane as 'h'.
From the top of the building, the angle of elevation to the airplane is 31°. This means that the tangent of this angle is equal to the height of the airplane divided by the distance between the top of the building and the airplane.
Using the tangent function, we can write:
tan(31°) = h / d
where 'd' is the distance between the top of the building and the airplane.
Similarly, from the base of the building, the angle of elevation to the airplane is 40°. Using the same logic, we can write:
tan(40°) = h / (d + 40)
Notice that we added the height of the building (40 m) to the distance 'd' to account for the base measurement.
Now we have a system of two equations:
tan(31°) = h / d
tan(40°) = h / (d + 40)
We can solve this system of equations to find the value of 'h'.
First, let's solve the first equation for 'd'. Multiply both sides by 'd':
d * tan(31°) = h
Next, let's substitute this expression for 'd' in the second equation:
tan(40°) = h / (d + 40)
tan(40°) = h / (h / tan(31°) + 40)
Now let's solve this equation for 'h'. Multiply both sides by (h / tan(31°) + 40):
tan(40°) * (h / tan(31°) + 40) = h
Now we can substitute the known values (angle measures) and use a calculator to solve for 'h':
tan(40°) * (h / tan(31°) + 40) = h
Replace the tangent values:
0.8391 * (h / 0.6009 + 40) = h
Distribute the multiplication:
0.8391 * h / 0.6009 + 0.8391 * 40 = h
Multiply both sides by the denominator:
0.8391 * h + (0.8391 * 40 * 0.6009) = h
0.8391 * h + 20.1686 = h
Subtract 0.8391 * h from both sides:
20.1686 = h - 0.8391 * h
Combine like terms:
20.1686 = 0.1609 * h
Divide both sides by 0.1609:
h = 20.1686 / 0.1609
Calculate:
h ≈ 125.38
Therefore, the altitude of the airplane is approximately 125.38 meters.