meg rowed her boat upstream a distance of 33 mi and then rowed back to the starting point the total time of the trip was 14 hours if the rate of the current was

Question

meg rowed her boat upstream a distance of 33 mi and then rowed back to the starting point the total time of the trip was 14 hours if the rate of the current was 4 mph find the average speed of the boat relative to the water?

Answer 1

Britteny,

 

Let's assume the speed of the boat in still water is constant at v mph.

 

Then the speed on the upstream leg is v-4 and downstream v+4 mph.

 

Time upstream leg=33/(v-4)

 

Time downstream leg=33/(v+4)

 

Total time=14=66v/(v^2-16) obtained by summing the two fractions above.

 

So 14v^2-66v-224=0       or    7v^2-33v-112=0

 

Using quadratic formula:  v=(33+/-sqrt(33^2+4*7*112))/14

 

Taking the positive root: v=(33+sqrt(4225))/14=98/14=7

 

So the constant speed of the boat relative to the water is 7 mph.

 

Note:  I am not convinced that if the boat has a variable speed that its "average" speed will be 7 mph (in fact I am sure it isn't) so beware of the use of that word.  "Constant" is more precise.

 

Check:  Speed upstream=3 mph taking 11 hours to go 33 miles.   Speed downstream=11 mph taking 3 hours.  Total time is 14 hours OK.