Question 19130: Hi, I'm stuck on the following problem: On a sales trip, Gail drive 600 miles to Richmond averaging a certain speed. The return trip is made at an average speed that is 10 miles per hour slower. Total time for the round trip is 22 hours. Find Gail's average speed on each part of the trip.
Any help would be appreciated. Thank you so much!!
Answer by mmm4444bot(95)   (Show Source):
You can put this solution on YOUR website! Hello There:
Hopefully, you've memorized the relationship between distance traveled, rate of speed, and elapsed time:
Distance = Rate * Time
(The asterisk means multiplication.)
Clearly, the distance going and coming is the same: 600
Let t1 and r1 represent the unknown elapsed time and rate going.
Let t2 and r2 represent the unknown elapsed time and rate coming back.
Since the problem gives us an amount for the total time, let's solve the equation above for time. Divide both sides by Rate to get:
Time = Distance/Rate
The rate coming back (r2) can be expressed in terms of the rate going (r1) because we're told that r2 is 10 less than r1.
r2 = r1 - 10
So, our two equations for elapsed time are:
t1 = 600/r1
t2 = 600/(r1 - 10)
The sum of these two times is 22.
600/r1 + 600/(r1 - 10) = 22
To simplify the appearance of the rest of the work, I'm going to type r instead of r1 from here on out.
600/r + 600/(r - 10) = 22
We can clear the fractions by multiplying both sides of this equation by the common denominator: r*(r - 10).
600*(r - 10) + 600*r = 22*r*(r - 10)
Carry out the multiplications (the distributive property) to get rid of the parentheses.
600*r - 6000 + 600*r = 22*r^2 - 220*r
Combine like terms, and move everything to the left side by subtracting the terms on the right from both sides.
1200*r - 6000 - 22*r^2 + 220*r
-22*r^2 + 1420*r - 6000 = 0
This is a quadratic equation. All of the coefficients are even numbers, so we can make them smaller by dividing both sides by 2.
-11*r^2 + 710*r + 3000 = 0
Use the quadratic formula to solve for r.
~ Mark
P.S. You should get 60 mph going and 50 mph coming back.
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