Systems of Equations

Question

A motorboat maintained a constant speed of 14 miles per hour relative to the water in going 21 miles upstream and then returning. The total time for the trip was 4.0 hours. Use this information to find the speed of the current.

Answer 1

This boat travels at 14 miles per hour in still water.

So it goes faster with a current pushing it and slower when the current is holding it back.

Lets let C equal the speed of the current.

So when the boat is going WITH the current, its Rate is 14 + C

And when the boat is going AGAINST the current, its Rate is 14 - C

The formula for distance is

Distance = Rate X Time

If you divide both sides of this equation by rate, you will get that

Time =. Distance

Rate

So the Time for going With the current is 21

14 + C

And the Time for going Against the current is 21

14 - C

Since the total time is 4 hours, we can add the time downstream and the time upstream to get 4.

So 21 + 21 = 4

14 + C 14 - C

Solving this equation for C will give the speed of the current.

So multiply each side of the equation by the common denominator (14 + C) (14 - C). Don't forget to multiply the 4 by that value also. It should look like this:

21 (14 + C) (14 - C) +. 21 (14 + C) (14 - C) = 4 (14 + C)(14 - C)

(14 + C) (14 - C)

Cancelling the common factors in each fraction gives

21 (14 + C) (14 - C) + 21 (14 + C) (14 - C) = 4 (14 + C)(14 - C)

(14 + C) (14 - C)

This simplifies to

21 (14 - C) + 21 (14 + C) = 4 ( 196 - C² ) Remember that (14 + C)(14 - C) = 14² - C²

Continue expanding and collecting like terms gives

294 - 21 C + 294 - 21 C = 784 - 4 C²

588 = 784 - 4 C²

4 C² = 784 - 588 = 196 Solving for C²

Dividing by 4 gives

C² = 196 / 4 = 49 Taking the square root of both sides.

C = 7

The current is traveling at 7 miles per hour.

Lets see if that makes sense.

So going WITH the Current, the boat is traveling at 14 + 7 or 21 miles per hour.

And going AGAINST the Current, the boat is traveling at 14 - 7 or 7 miles per hour.

21 miles of distance at 21 miles per hour will take 1 hour.

21 miles of distance at 7 miles per hour will take 3 hours.

That is a total of 4 hours round trip. IT CHECKS!!

Answer 2

If s = the speed of the current (that we're solving for), then

the speed of the boat going upstream is 14 – s

the speed of the boat going upstream is 14 + s

The relationship between distance (d), time (t) and speed (s) is

d = s × t which means that t = d/s

We know that the total time for the trip is 4 hrs

The time it takes to go upstream (t_u) is

t_u = 21/(14 – s)

The time it takes to go upstream (t_d) is

t_d = 21/(14 + s)

and

t_u + t_d = 4

So the equation we use to solve for the speed of the current (s) is

21/(14 – s) + 21/(14 + s) = 4

Multiply both sides by (14 – s)(14 + s) to get

21(14 + s) + 21(14 – s) = 4(14 – s)(14 + s)

Expanding the LHS and noting that (14 – s)(14 + s) = (14)² – s² = 196 – s² we get

294 + 21s + 294 – 21s = 588 = 4(196 – s²) = 784 – 4s²

which simplifies to

588 = 784 – 4s² which means that

4s² = 784 – 588 = 196

Divide both sides by 4 to get

s² = 196/4

and

s = √(196/4) gives you the speed of the current.