speed question

Question

a motorboat moves 15 miles per hour in still water. It travels 27 miles upstream and 27 miles downstream in a total time of 10.0 hours. what is the speed of the current?

Answer 1

Letting c = speed of the current:

Upstream is against the current so speed upstream is 15 - c

Downstream is with the current so speed downstream is 15+c

 

If he travels h hours upstream, then he travels 10-h hours back downstream

 

Distance formula:  distance = rate*time ...  d= rt

 

27 = (15-c)h         equation 1

27 = (15+c)(10-h)   equation 2

 

(15-c)h = (15+c)(10-h)

15h - ch = 150 - 15h + 10c - ch

15h = 150 - 15h + 10c

30h = 150 + 10c

 

10c = 30h - 150

c = 3h - 15

 

27 = (15-c)h     from equation 1

27 = (15-(3h-15))h

27 = (15-3h +15)h

27 = (30-3h)h

27 = 30h - 3h²

 

3h² - 30h + 27 = 0

h² - 10h + 9 = 0

(h-9)(h-1) = 0

h = 9 hours upstream

10 - h = 1 hour downstream

 

he travels 9 hours upstream and 1 hour downstream

 

c = 3h-15 so:

 

c = 3(9)-15 = 27-15 = 12 mph