Chapter 4 | Applications of Derivatives
343
The volume of a sphere of radius r centimeters is V = 4 3 πr 3 cm 3 . Since the balloon is being filled with air, both the volume and the radius are functions of time. Therefore, t seconds after beginning to fill the balloon with air, the volume of air in the balloon is V ( t ) = 4 3 π [ r ( t )] 3 cm 3 . Differentiating both sides of this equation with respect to time and applying the chain rule, we see that the rate of change in the volume is related to the rate of change in the radius by the equation V ′( t ) =4 π ⎡ ⎣ r ( t ) ⎤ ⎦ 2 r ′( t ). The balloon is being filled with air at the constant rate of 2 cm 3 /sec, so V ′( t ) =2cm 3 /sec. Therefore, 2cm 3 /sec= ⎛ ⎝ 4 π ⎡ ⎣ r ( t ) ⎤ ⎦ 2 cm 2 ⎞ ⎠ · ⎛ ⎝ r ′( t )cm/s ⎞ ⎠ , which implies r ′( t ) = 1 2 π ⎡ ⎣ r ( t ) ⎤ ⎦ 2 cm/sec. When the radius r =3cm, r ′( t ) = 1 18 π cm/sec.
What is the instantaneous rate of change of the radius when r =6cm?
4.1
Before looking at other examples, let’s outline the problem-solving strategy we will be using to solve related-rates problems.
Problem-Solving Strategy: Solving a Related-Rates Problem 1. Assign symbols to all variables involved in the problem. Draw a figure if applicable. 2. State, in terms of the variables, the information that is given and the rate to be determined. 3. Find an equation relating the variables introduced in step 1. 4. Using the chain rule, differentiate both sides of the equation found in step 3 with respect to the independent variable. This new equation will relate the derivatives. 5. Substitute all known values into the equation from step 4, then solve for the unknown rate of change.
Note that when solving a related-rates problem, it is crucial not to substitute known values too soon. For example, if the value for a changing quantity is substituted into an equation before both sides of the equation are differentiated, then that quantity will behave as a constant and its derivative will not appear in the new equation found in step 4. We examine this
potential error in the following example. Examples of the Process
Let’s now implement the strategy just described to solve several related-rates problems. The first example involves a plane flying overhead. The relationship we are studying is between the speed of the plane and the rate at which the distance between the plane and a person on the ground is changing.
Made with FlippingBook - professional solution for displaying marketing and sales documents online