Question

From a tower 150 feet high, with the sun directly overhead, and airplane and its shadow have an angle of elevation of 67.4° and an angle of depression of 8.1°, respectively. Find the height of the airplane.

Please show all steps and final answer.

Answer 1

To find the height of the airplane, we can use trigonometry and the angles of depression and elevation. By using the tangent function and the given height of the tower, we can calculate the height of the airplane as approximately 562.21 feet.

Step-by-step explanation:

To find the height of the airplane, we can use trigonometry. Let's start by considering the angle of depression. The angle of depression is the angle between the horizontal plane and the line of sight from the observer to the object below the observer.

Given that the angle of depression is 8.1° and the height of the tower is 150 feet, we can use the tangent function to find the distance from the tower to the airplane's shadow. Tan(8.1°) = shadow distance / 150.

The angle of elevation is the angle between the horizontal plane and the line of sight from the observer to the object above the observer. The angle of elevation is complementary to the angle of depression, so it is 90° - 8.1° = 81.9°.

Now we can use the height of the tower and the tangent function to find the height of the airplane. Tan(81.9°) = airplane height / 150.

Now we can solve for the height of the airplane: Tan(81.9°) * 150 = airplane height. Calculating this, we find that the height of the airplane is approximately 562.21 feet.