You observe a plane approaching overhead and assume that its speed is 700 miles per hour. The angle of elevation of the plane is 16° at one time and 58° one minute later. Approximate the altitude of the plane
During takeoff, an airplane's angle of ascent is 16° and its speed is 300 feet per second.
(a) Find the plane's altitude after 1 minute. (Round your answer to the nearest whole number.)
ft
(b) How long will it take the plane to climb to an altitude of 10,000 feet? (Round your answer to one decimal place.)
sec
To approximate the altitude of the plane, we can use the concept of trigonometry and the information provided.
First, let's define some variables:
- Let "A" be the altitude of the plane.
- Let "d" be the distance traveled by the plane in one minute.
Since the plane is assumed to be moving at a speed of 700 miles per hour, we can calculate the distance traveled by the plane in one minute as follows:
d = (700 miles / 60 minutes) = 11.67 miles.
Now, let's consider the right-angled triangle formed by the observer, the plane, and its altitude. The angle of elevation changes from 16° to 58°, which means we have two similar right-angled triangles.
Using the tangent function, we can relate the angle of elevation to the altitude and the distance traveled by the plane:
tan(16°) = A / d
Solving for A, we get:
A = d * tan(16°)
Substituting the value of d, we have:
A = 11.67 miles * tan(16°)
Using a scientific calculator or trigonometric table, we can find that tan(16°) is approximately 0.29191.
Therefore, the approximate altitude of the plane is:
A ≈ 11.67 miles * 0.29191 ≈ 3.402 miles.
So, the altitude of the plane is approximately 3.402 miles.