Question 963427: Suzanne rides the chairlift to the top of a 4 mile hill, then skis down at 8 miles per hour faster than lift travels. The trip is 8 minutes faster than the ride up. Find her average speed for the entire trip.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
R is equal to the rate of speed on the way up.
R + 8 is equal to the rate of speed on the way down.
T is equal to the time it takes to get to the top from the bottom.
T - 8/60 is equal to the the time it takes to get to the bottom from the top.
you needed to convert everything to either minutes or hours.
i chose hours.
8/60 hours is equal to 8 minutes.
since rate * time is equal to distance, then:
R*T = 4 on the way up.
(R+8) * (T - 8/60) = 4 on the way down.
you solve each of these equations for R to get:
R = 4/T from the equation for going up.
R = 4/(T-8/60) - 8 from the equation for coming down.
since both expressions on the right side of the equation are equal to R, you can set them equal to each other to get:
4/T = (4/(T-8/60)) - 8
you then solve for T to get T = 1/3 of an hour.
the solving results in a quadratic equation of 480T^2 - 64T - 32 = 0 which can then be factored to get 15T^2 - 2T - 1 = 0 which can then be factored to get (5T+1) * (3T-1) = 0 which can then be solved for T to get T = -1/5 or T = 1/3.
Since T can't be minus, the only possible solution is T = 1/3.
once you've done that, you then solve for R to get R = 12.
For the ride up, R * T = 4 becomes 1/3 * 12 = 4 miles.
For the ride down, (R+8) * (T - 8/60) becomes (12+8) * (1/3 - 8/60) which becomes 20 * (20/60 - 8/60) which becomes 20 * 12/60 which becomes 20 * 1/5 which becomes 20/5 which becomes 4.
T = 1/3 of an hour and R = 12 miles per hour is confirmed to be good.
the total time it took to go up and down is 1/3 of an hour to go up + 12/60 of an hour to go down which makes the total time equal to 1/3 + 12/60 = 20/60 + 12/60 = 32/60 = 8/15 of an hour.
the average speed for the total trip up and down is therefore 8 / (8/15) which is equal to 8 * 15/8 which is equal to 15 miles per hour.
this is derived from the rate * time = distance formula again.
distance round trip is 8.
time round trip is 8/15 of an hour.
R * T = 8 becomes R * 8/15 = 8 which becomes R = 8 * 15/8 which becomes R = 15.
|
|
|
