SOLUTION: a motorboat heads upstream for a distance of 24 miles on the Illinois River, whose current is running at 3mph. the trip up and back takes a total of 6 hours. assuming that the boat

Question 167652: a motorboat heads upstream for a distance of 24 miles on the Illinois River, whose current is running at 3mph. the trip up and back takes a total of 6 hours. assuming that the boat maintained a constant speed relative to the water, what is this constant speed? -- i've never been great at calculus word problems, thank u for all the help.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Downstream:

Start with the distance-rate-time formula

Plug in d=24

Replace with (since the speed of the river sped up the boat 3 mph)

Divide both sides by and rearrange to isolate "t"

So the time it took to go downstream was

------------------------

Upstream:

Start with the distance-rate-time formula

Plug in d=24

Replace with (since the speed of the river slowed the boat 3 mph)

Divide both sides by and rearrange to isolate "t"

So the time it took to go upstream was

-------------------------

Now simply add the two times and set them equal to the total time (6 hours) to get

Downstream Time + Upstream Time = Total Time

-------------------------

Start with the given equation

Multiply EVERY term by the LCD to clear the fractions

Multiply and simplify

FOIL

Distribute

Combine like terms.

Subtract 48r from both sides

Rearrange the terms.

Notice we have a quadratic equation in the form of where , , and

Let's use the quadratic formula to solve for r

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Rewrite as

Add to to get

Multiply and to get .

Take the square root of to get .

or Break up the expression.

or Combine like terms.

or Simplify.

So the possible answers are or

However, a negative speed doesn't make much sense. So the only answer is

===================================

Answer:

So the solution is which means that the boat was going 9 mph relative to the water.

In other words, the speed of the boat in still water is 9 mph.
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