Laboratory 6
Related Rates

Introduction

Related rates problems arise when one knows the rate of change of one particular quantity, and would like to discover the rate of change of some second quantity which is related to the first.

6.1  Example

As an example, consider the following problem.

An airplane at an altitude of 3 kilometers is flying at a speed of 400 kilometers per hour on a flight path that will take it directly over a radar antenna. How fast is the plane approaching the radar antenna when the plane is 8 kilometers away on ground from the radar antenna?

We proceed with the following guidelines for solving related rate problems.

1.  Determine two quantities that are changing with time. Make a sketch by hand and label  the quantities.

Let:		x represent the ground distance that the plane has traveled; and
		s represent the distance between the plane and the radar antenna.

The sketch, in this case looks like the following one.

2.  We then identify all given quantities as well as the quantities to be determined.

               	Given quantities: 		d x/d t=-400
               	Quantities to be determined: 	d s/d t when x=8.

3.  Next, we find an equation that relates the two quantities. Note that these quantities are changing with time.  In the above case, we note that both s and x are function of time t and the relation between them is ((s(t))^2)=((x(t))^2)+(3^2).

4.  We express the equation using Mathematica replacing s and x by s(t) and x(t) because they are functions of t.

Clear[eq,s,x,t]

eq=(s[t]^2==x[t]^2+3^2)

Note that, in the above, == is used for denoting equality in the equation.

5.  We differentiate both sides of the equation with respect to time  t.

Clear[derivativeeq]

derivativeeq=D[eq,t]

6.  We solve  the resulting equation for s'[t]  because we want s'[t] when x = 8. We perform this operation with the Mathematica command Solve. We then assign the name srate to the result.

srate=Solve[derivativeeq,s'[t]]

7.  We calculate the numerical value of s(t) when x(t)=8 using the equation that we found in Step 3 above. Note that the values of x(t) and x'(t) are known, namely x'(t)=-400 and x(t)=8 at a particular instant of time. In order to compute s'(t) we only need s(t) at this particular instant of time.

Solve[s^2==9+8^2,s]

8.  We find the answer s'[t] by substituting the values x'(t) =-400, x(t)=8, and s(t)=(Sqrt[73]).

theanswer=srate /. {x'[t]->-400, x[t]->8, s[t]->Sqrt[73]}

N[theanswer]

Hence, the plane is approaching (note the minus sign in front) the radar antenna at the speed of 374.532 kilometers per hour when the plane is 8 kilometers away on the ground from the radar antenna.

Experiment 6.1a

Let V(t) represent the volume of a spherical weather balloon at time t and r(t) represent the radius of the balloon at time t. Helium is being pumped into the balloon at a rate of  500 cubic centimeters per second. At what rate is the radius of the balloon increasing when the diameter of the balloon is 200 centimeters?

Experiment 6.1b

A boat is pulled in by means of a winch on the dock 12 feet above the deck of the boat. The winch pulls in the rope at the rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock?

Credits

These laboratories were created by Raouf Boules, Geoff Goodson, Ohoe Kim and Mike O'Leary for use in the Calculus courses of Towson University. Commercial use is prohibited without permission of the authors. Non-commercial use is permitted, provided this credit section is retained.