Question

Flight Control An airplane is flying in still air with an airspeed of 240 miles per hour. If it is climbing at an angle of \(22^{\circ},\) find the rate at which it is gaining altitude.

Short answer

The airplane is gaining altitude at a rate of approximately 89.9 miles per hour.

Step-by-step solution

Understand the problem

The plane's airspeed (240 miles per hour) tells us how fast the plane is moving through the air, but it doesn't tell us how fast the plane is ascending. To find that, we need to break down the plane's motion into horizontal and vertical components using the angle of ascent (22 degrees).

Use the sine function to find the vertical speed

We can use one of the basic functions of trigonometry: the sine. The sine of an angle, in a right triangle, is the ratio of the length of the side opposite the angle (the 'opposite' side) to the length of the hypotenuse. Here, the hypotenuse is the airspeed, and the 'opposite' side is the vertical speed we want to find. Therefore, we can set up the equation: the sine of 22 degrees is equal to the vertical speed divided by the airspeed. From this, we can solve for the vertical speed by multiplying both sides of the equation by the airspeed.

Calculate the value

The sine of 22 degrees is approximately 0.3746 (using a calculator). Therefore, the vertical speed is approximately 0.3746 times the airspeed, which is 240 miles per hour. Therefore, the vertical speed or the rate at which the airplane is gaining altitude is approximately 89.9 miles per hour.