A boat traveled downstream a distance of 120 mi and then came right back. If the speed of the current was 7 mph and the total trip took 5 hours, find the averag

Question

A boat traveled downstream a distance of 120 mi and then came right back. If the speed of the current was 7 mph and the total trip took 5 hours, find the average speed of the boat relative to the water?

Answer 1

Hi Britteny!

 

So lets think about this problem in two parts - upstream and downstream. We have traveled a total of 240 miles in 5 hours.

 

Upstream our speed will be 7mph slower since we are moving against the current.

Downstream our speed will be 7mph faster since we are moving with the current.

 

So!

 

 

240mi / 5hr = 48mi/hr = 48mph

 

 

This little equation gives us our average speed for the entire trip (total distance / total time). Now we factor in the current, and we get 41mph upstream and 55mph downstream. There's our answer! Just to doublecheck though:

 

120mi / 41mph = 2.9 hours

 

120mi / 55mph = 2.1 hours

 

2.1 + 2.9 = 5.0

 

So our answers are consistent with the problem and we have our answers!

Answer 2

So let x be the actual speed of the boat. When traveling downstream, the current and the boat is in the same direction and their speeds add so the boat travels at an apparent speed of x+7 mph. Traveling upstream is the opposite, current and boat work against each other and the boat travels at an apparent speed of x-7 mph. 

 

Let's use those 2 terms in equations to figure out the time the boat takes to travel each way:

 

Downstream: 120mi = (x+7)*t_d ---> t_d = 120/(x+7)

Upstream: 120mi = (x-7)*t_u ---> t_u = 120/(x-7)

 

In total: t_d+t_u = 5hrs 

 

Substituting:

 

120/(x+7) + 120/(x-7) = 5

 

Simplifying ultimately leads to the quadratic: x²-48x-49=0

 

Factoring give you: (x-49)(x+1)=0 ---> x=49 or -1

 

Since speed is a positive number, the boat is going at 49mph relative to the water. 

 

To check your answer:

 

Going downstream: 49+7 = 56mph ---> 120mi/56mph = ~2.143hr

Going upstream: 49-7 = 42mph ---> 120mi/42mph = ~2.857hr

 

2.143hr + 2.857hr = 5hr