Objectives 

• Identifying the Relevant Variables

• Constructing Equations 

• Solving Equations and Checking Solutions

Examples

• Example 1

• Example 2

• Example 3

• Example 4

A boat travels upstream (against the current) and then returns home. The speed of the boat's engine in still water is 15 mph. The trip upstream lasts 45 minutes, and the trip back downstream lasts 30 minutes. What is the speed of the current in miles per hour?

Hint: Let  x = speed of the current in miles per hour
 

Solution
    Our goal is to construct a one-variable equation which 
    matches the given information.

    Step 1: Identify the desired variable.

        The hint tells us to use the variable "x" to represent the
        speed of the current in miles per hour.

        That is,

x = speed of the current in miles per hour

    Step 2: Identify the known relationships in the problem.

        The problem statement tells us that boat has a constant
        motor speed of 15 mph.

        Note: As the boat travels upstream, the speed of the
                   current will "take away from" the speed of the boat.

                    We can translate this fact as follows:

upstream boat speed = 15 - x

                    As the boat travels downstream, the speed of the
                    current will "add to" the speed of the boat.

               We can translate this fact as follows:

downstream boat speed = 15 + x

        We will also need the formula:

(rate)(time) = distance

        Note:  In this formula, we need matching units.

        Since the rates are given in miles per hour, we need to
        convert the times to hours.

        That is,

 45 minutes = 0.75 hours

 30 minutes = 0.5 hours

        We can construct a table to help us organize the given
        information.
 

	rate (mph)	time  (hr)	distance (miles)
upstream	15 -  x	0.75	(0.75)(15 - x)
downstream	15 + x	0.5	(0.5)(15 + x)

Step 3: Construct a one variable equation which fits the known
              relationships in the problem.

       In this problem, the trip upstream and the trip downstream 
       cover the same distance.

       Using the entries from the "distance" column in our
       table, we have the equation:

upstream distance = downstream distance

(0.75)(15 - x) = (0.5)(15 + x)

Step 4: Solve the equation.

        Solving for x, we get

11.25 - 0.75x = 7.5 +0.5x

 - 1.25x = - 3.75

x = ( - 3.75)/( - 1.25) = 3

Step 5: Use the solution from Step 4 to answer the question(s)
              posed in the original problem. 

The speed of the current is 3 mph.


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Special Note: Be sure to check that your final answers "make sense" in the context of the original problem. 

In this problem, for example, our equation needs to yield a non-negative number for the value of "x". 

That is, it wouldn't "make sense" for the speed of the current to be - 3 mph. And if the speed of the current equaled zero, the upstream time and the downstream time would have to be equal.