To the nearest minute, what time did Grant and Sadie arrive at Elk island while on their rowing adventure?

Question

Grant and Sadie are experienced rowers. One day, they hopped in a row boat and took off upstream on a river, where the current was a constant 3 miles per hour. They began this rowing adventure at 8:30 am. When they finally got to Elk island, the two towers deicded to take a lunch break of one and a half hours. After lunch, Grant and Sadie paddled back downstream and finally arrived back to their starting point at 4:00 pm. Grant and Sadie traveled a total of 24 miles on this trip. It is OK to assume that Grant and Sadie always paddle in such a way that they would be traveling the same speed in still water. To the nearest minute, what time did Grant and Sadie arrive at Elk island while on their rowing adventure?

Answer 1

Hi Luke,

 

This is a fun problem that requires some thinking. . .

 

They began at 8:30, rested for 1 1/2 hours and got back at 4:00.  That means that their rowing time was 6 hours.

 

They traveled a total of 24 miles . . . that means that they rowed 12 miles upstream and then 12 miles downstream.

 

Let X = their rowing speed in still water

 

Then they traveled at a rate of X -3 miles per hour upstream because the current slowed them down by 3 mph

 

They traveled at a rate of X + 3 miles per hour downstream because the current gave them a push by 3 mph

 

The formula is distance/rate = time.  The time up is 12/(X-3) and the time back is 12/(X+3) and the total time is 6 hours:

 

12/(X - 3) + 12/ (X + 3) = 6

 

multiply first fraction by (X+3)/(X+3) and the second fraction by (X-3)/(X-3) to get a common denominator:

 

(12(X+3) + 12 (X-3))/ (X+3)(X-3) = 6

 

(12X + 36 +12X - 36)/ (X^2 -9) = 6

 

24X / (X^2 -9) = 6

 

Cross multiply--> 24X = 6X^2 - 54

 

Move everything to the right side of the equation-->0 =  6X^2 - 24X - 54

 

Divide everything by 6 --> x^2 - 4X - 9 = 0

 

Use the quadratic equation with a = 1   b = -4    c = -9

 

X = (4 +/- sq rt (16 + 36) / 2

 

X = (4 +/- sq rt 52) / 2

 

X = (4 +/- 7.2) / 2

 

X = 5.6 miles per hour  (we can disregard the negative value of X)

 

This means that they traveled 2.6 mph upstream (by subtracting 3 mph from 5.6 mph)

 

The distance is 12 miles

 

Now you can calculate the time to Elk Island with the formula distance/rate = time and then figure what time they arrived at Elk Island since they began at 8:30 AM

 

Have fun.

 

Dan