SOLUTION: 4. The Hudson River flows at a rate of 5 miles per hour. A patrol boat travels 40 miles upriver, and returns in a total time of 6 hours. What is the speed of the boat in still wa

Question 180490: 4. The Hudson River flows at a rate of 5 miles per hour. A patrol boat travels 40 miles upriver, and returns in a total time of 6 hours. What is the speed of the boat in still water?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

This is a distance, rate, and time problem, so the basic formula to use is

. However, we will find it convenient to express this relationship as

for this particular problem.

We are asked to find the rate of the boat in still water, so let's call that

to distinguish it from the r in the basic formula above. We are given the speed of the current, 5 mph, so we know that the speed of the boat relative to the shoreline as the boat goes upstream is

and the speed of the boat relative to the shoreline as the boat goes downstream is

. We also know that each leg of the trip is 40 miles long. Lastly, let's say that the time it takes for the boat to make the upstream trip is t hours. That means that, since the entire trip took 6 hours, the downstream trip must have taken 6 - t hours.

Now let's use the formula,

to describe the upstream trip:

And the downstream trip:

First, solve the downstream equation for t:

And simplify the result:

Next if you compare the simplified form of the downstream equation with the upstream equation, you will notice that we have two different expressions in

that are both equal to the same thing, namely t. We can now set these two expressions equal to each other:

Put everything on the left and apply the common denominator

Multiply by

Apply the distributive property and collect terms:

Multiply by

Factor:

Therefore

The negative answer is absurd because we know the boat wasn't going backwards, so exclude that answer as an extraneous root introduced by the action of squaring the variable during the solution process, and that leaves us with

Check the answer.

If the speed in still water is 15 mph, then the upstream speed must have been 15 - 5 = 10 mph. At 10 mph, it takes

hours to make the trip. The downstream speed must have been 15 + 5 = 20 mph. At 20 mph, it takes

hours to make the trip. 4 hours plus 2 hours = 6 hours which was the given elapsed time. Answer checks.

John

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