Motion word problem?

Question

In still water, Sarah's boat can tavel 15km/h. if it takes her a total of 4.5 h to travel 30 km up a river and then to return by the same route, what is the speed of the current in the river?

Can you please explain this to me, using a mathematical equation?

Answer 1

Quite simple really

x = distance
v=speed
t=time (to go up river)
c=speed of current

x=vt

30 = (15 - c)t
30 = (15 + c)(4.5-t)

t = 30/(15-c)
30 = (15 + c)(4.5 - (30/(15 - c)))
30(15 - c) = (15 + c)(4.5(15-c)-30)

450 - 30c = (15 + c)(37.5-4.5c)
900-60c=(15+c)(75-9c)
900-60c=1125-135c+75c-9c^2
9c^2-225=0
(3c+15)(3c-15)=0

c=5

Answer 2

Let the river speed be v km/h.

When she goes downstream, the river helps her so her speed (relative to a person standing on the bank and watching her) becomes v+15 in km/h. While upstream it is 15-v.

Time to go upstream = Distance/speed = 30/(v-15)
Time to go downstream = Distance/speed = 30/(v+15)

We are told that total time = 4.5 h = 30/(v-15)+ 30/(v+15).

Multiply both sides by (2/3)(v+15)(v-15), which is actually equal to v*v-225:

3v^2-675=20(v+15) + 20(v-15)
or, 3v^2-675 = 40v

Solve this quadtratic equation to get v.