A truck travels from X to Y.
When the truck is going uphill, it goes at 56 mph, when it's going downhill, it goes at 72 mph, and when the ground is flat it goes at 63 mph. Given that it takes the truck 4 hours to get from X to Y and 5 hours to get from Y to X, what is the distance between X and Y?
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I really don't know where to start with this - can someone please help?
With very many thanks,
Froskoy.
MathMan TG
Favorite Answer
When you reverse the direction,
uphill becomes downhill, downhill becomes uphill, and flat stays flat.
Suppose when going from x to y we have
u miles of uphill,
f miles of flat
d miles of downhill
(and the reverse coming back).
x to y: u / 56 + f / 63 + d / 72 = 4 hours
y to x: u / 72 + f / 63 + d / 56 = 5 hours
Subtracting top from bottom:
u/72 - u/56 + d/56 - d/72 = 1
1/72 - 1/56 = 1/8*9 - 1/7*8 = -2/7*8*9
-2/7*8*9 u + 2/7*8*9 d = 1
-2u + 2 d = 7*8*9 = 504
2 (d - u) = 504
d - u = 252
And we want
u/56 + f/63 + d/72 = 4
u/72 + f/63 + d/56 = 5
Now suppose u = 0.
Then we get d = 252
x to y takes: 0/56 + f/63 + 252/72 = 3.5 + f/63 hours
f/63 = 0.5 hour (to bring total to 4)
f = 31.5
and total distance = 252 = 283.5
For the return journey we have
0/72 + 0.5 + 252/56 = 0.5 + 4.5 = 5 hours, as required.
Now we can repeat this calculation for any value of u
that we like up to and including 15.75,
which increases d as well, but reduces f.
When u = 15.75, f becomes 0, so that is the maximum value for u.
(and d = 267.75, and we have
15.75 / 56 + 267.75 / 72 = .28125 hours uphill + 3.71875 hours downhill, total 4
and 15.75 / 72 + 267.75 / 72 = .21875 hours downhill + 4.78125 hours uphill, total 5,
on the reverse trip).
But in every case,
u + f + d = 283.5 miles, and that is the answer to the problem.