344
Chapter 4 | Applications of Derivatives
Example 4.2 An Airplane Flying at a Constant Elevation
An airplane is flying overhead at a constant elevation of 4000ft. A man is viewing the plane from a position 3000ft from the base of a radio tower. The airplane is flying horizontally away from the man. If the plane is flying at the rate of 600 ft/sec, at what rate is the distance between the man and the plane increasing when the plane passes over the radio tower?
Solution Step 1. Draw a picture, introducing variables to represent the different quantities involved.
Figure 4.3 An airplane is flying at a constant height of 4000 ft. The distance between the person and the airplane and the person and the place on the ground directly below the airplane are changing. We denote those quantities with the variables s and x , respectively.
As shown, x denotes the distance between the man and the position on the ground directly below the airplane. The variable s denotes the distance between the man and the plane. Note that both x and s are functions of time. We do not introduce a variable for the height of the plane because it remains at a constant elevation of 4000ft. Since an object’s height above the ground is measured as the shortest distance between the object and the ground, the line segment of length 4000 ft is perpendicular to the line segment of length x feet, creating a right triangle. Step 2. Since x denotes the horizontal distance between the man and the point on the ground below the plane, dx / dt represents the speed of the plane. We are told the speed of the plane is 600 ft/sec. Therefore, dx dt =600 ft/sec. Since we are asked to find the rate of change in the distance between the man and the plane when the plane is directly above the radio tower, we need to find ds / dt when x =3000ft. Step 3. From the figure, we can use the Pythagorean theorem to write an equation relating x and s : ⎡ ⎣ x ( t ) ⎤ ⎦ 2 +4000 2 = ⎡ ⎣ s ( t ) ⎤ ⎦ 2 . Step 4. Differentiating this equation with respect to time and using the fact that the derivative of a constant is zero, we arrive at the equation
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online