System of Equations

Question

A motorboat can maintain a constant speed of 37 miles per hour relative to the water. The boat makes a trip upstream to a certain point in 39 ​minutes; the return trip takes 35 minutes. What is the speed of the​ current?

Answer 1

We do not know the speed of the current.

Let vc = x be this speed.

This problem involves distance, time and speed, so we will use the formula D = VT, where

D = distance

V = speed

T = time

Summarize the givens:

Speed of the boat: vb = 37 miles per hour

Speed of the water: vc = x miles per hour

Time upstream: t1 = 39 minutes = 39/60 hours

Time downstream: t2 = 35 minutes = 35/60 hours

Distance is same both ways: d1 = d2

When the boat goes upstream, it will travel slower because it is going against the current, the overall speed will be:

v1 = vb - vc = 37 - x

When the boat goes downstream, it will travel faster because it is going with the current, the overall speed will be:

v2 = vb - vc = 37 + x

Since the boat travels downstream and upstream the same distance, we know that

d1 = d1

v1t1 = v2t2

Substituting values into these formulas, we have

39/50(37 - x) = 35/60(37 + x) -- multiply both sides by 60

39(37 - x) = 35(37 + x) -- distribute on both sides

1443 - 39x = 1295 + 35x -- subtract 1295 from both sides

148 - 39x = 35x -- add 39x to both sides

148 = 74x -- divide both sides by 74

2 = x -- final answer

Therefore, the speed of the current is 2 miles per hour.

Answer 2

If the speed of the current is x mph then the resultant speed of the boat is (37-x) upstream and (37+x) downstream.

So if s is the length of the trip upstream, then s = (39/60) (37-x)

and downstream s = (35/60) (37 +x)

So (39/60) (37-x) = (35/60) (37+x)

39 (37 - x) = 35 (37 + x)

1443 - 39x = 1295 + 35x

148 = 74 x

x = 148 / 74 = 2 mph