Question

A light aircraft was traveling 108 miles per hour straight and level when the pilot measured a landmark at a∘4.3 degree angle of depression. After 47 minutes the aircraft was directly over the landmark. At what height, to the nearest foot, is the aircraft flying?

Answer 1

Using the angle of depression and trigonometry, the altitude of the light aircraft is calculated to be approximately 33,553 feet to the nearest foot.

Step-by-step explanation:

The student's question involves calculating the height at which the aircraft is flying using the angle of depression and the distance traveled. To solve this problem, we use trigonometry, specifically tangent function, which relates the angle of depression to the height of the aircraft and the distance covered.

Given the speed of the aircraft (108 miles per hour), the angle of depression (4.3 degrees), and the time taken to reach the landmark (47 minutes), we can calculate the distance the aircraft traveled and then find the altitude.

First, convert the speed to miles per minute:
108 miles per hour ÷ 60 minutes per hour = 1.8 miles per minute.
Calculate the distance traveled in 47 minutes:
1.8 miles per minute × 47 minutes = 84.6 miles.

Using the tangent function:
tan(4.3°) = opposite/adjacent = height/distance.
Therefore, the height can be calculated as follows:
height = tan(4.3°) × 84.6 miles

To get the height in feet, first convert miles to feet (1 mile = 5280 feet):
84.6 miles × 5280 feet/mile = 446,688 feet.
Then, calculate the height:
height = tan(4.3°) × 446,688 feet

Use a calculator to find tan(4.3°), which approximately equals 0.0751, then multiply this by the distance in feet:
0.0751 × 446,688 feet ≈ 33,553 feet.

To the nearest foot, the altitude of the aircraft is 33,553 feet.