SOLUTION: a motorboat can go 12 miles downstream on a river in 20 minutes for this boat ti go back upstream the same 12 miles. find the speed of the boat

Question 453410: a motorboat can go 12 miles downstream on a river in 20 minutes for this boat ti go back upstream the same 12 miles. find the speed of the boat
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
rate * time = distance

going against the current, the rate of the boat is (r - x) where r is the rate of the boat and x is the rate of the current.

going with the current, the rate of the boat is (r + x) where r is the rate of the boat and x is the rate of the current.

equation going downstream is (r+x) * time = distance.

equation going upstream is (r-x) * time = distance.

going downstream is with the current.

going upstream is against the current.

r = rate of the boat
x = rate of the current

example:

current is going downstream at 5 miles per hour.

boat takes 20 minutes to go downstream.

boat is going downstream for 12 miles.

before we do anything, we have to translate 20 minutes into 20/60 = .333333 hours so we are dealing with everything in hours.
equation for going downstream ( with the current) is (r + x) * time = distance.

equation for going upstream (against the current) is (r - x) * time = distance.

we have:

r = r
x = 5 miles per hour
time = .333333 hours
distance = 12 miles

going downstream, the equation is (r + 5) * time = distance

substituting what we know, the equation becomes:

(r + 5) * .333333 = 12

solving for r, we get:

r = 31 miles per hour.

the boat is traveling at 31 miles per hour.

substituting in the downstream equation of (x + r) * time = distance, we get:

(31 + 5) * .3333333 = 12

solving this equation, we get:

12 = 12 which confirms the value for r is good.

we can now use that number to see how long it takes the boat to go back upstream.

this time the boat is traveling against the current.

the equation for that is (r - x) * time = distance

this time we know what r is but we want to find time.

we have:

r = 31
x = 5
time = time
distance = 12

out equation of (r - x) * time = distance becomes:

(31 - 5) * time = 12 which becomes:

26 * time = 12 which becomes:

time = 12/26 = .4615384615 hours which can translate into minutes by multiplying by 60 to get:

time = 27.69230769 minutes.

this is approximately 27 minutes and 41 seconds.

going downstream takes 20 minutes.
going upstream takes 27 minutes and 41 seconds.
the speed of the boat is 31 miles per hour
the speed of the current is 5 miles per hour
the distance traveled is 12 miles going downstream and the same 12 miles going upstream.
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