Rates question?

Question

A car travels from town A to town B, a distance of 600km, in x hours. A plane, travelling 220km/h faster than the car, takes five and a half hours less to cover the same distance. Find the actual average speed of each of them.

Answer 1

Principle= speed x time = distance
Sooo Let velocity of Car= vc
velocity of plane=220+vc
Time for car to travel 600 km= 600/vc
Time for plane to travel= (600/vc)-5.5
So for plane, (220+vc)([600/vc]-5.5)= 600
This looks messy, and trial and error is the best way to solve without going into quadratic eqtns. vc= 80 km/hr, plane=300 km/hr

Answer 2

Its more correct to say that the car took 5.5 hours more. Because at 5.5 hours the plane would have arrived before ti left.

So now that we have the word problem re-worded to make sence.

The car took 5.5 hours longer.

First lets calculate the obvious, how fast was the plane going??

Velocity = Distance / Time
V = D / T

Vp = 600 / (Tcar - 5.5)

And how fast was the car going?

Vc = 600 / Tcar


Now at this point, since the problem asks for the average, you can simply add the two together, and divice by 2.

avg = (Vp + Vc)/2

or

AVG = (600 / (Tcar - 5.5) + 600 / Tcar) / 2 ... well it was worth a shot, but it looks like we need another equation to solve for Tcar, so lets combine the two.

Now, something is the same... So lets solve for it and write a new equation. Tcar is the same in both so, solve for Tcar

for the plane equation:
Vp = 600 / (Tcar - 5.5)
(Tcar - 5.5) = 600 / Vp

Tcar = (600 / Vp) + 5.5

Tcar = 600 / Vc

Now Equate


(600 / Vp) + 5.5 = 600 / Vc

And Solve for Vc

Vc = 600*((600 / Vp) + 5.5 )

Now we have a third equation for the velocity of the car, that can be subbed into the car equations.

Vc = 600 / Tcar

600*((600 / Vp) + 5.5 ) = 600 / Tcar
(600 / Vp) + 5.5 ) = 1 / Tcar

Now sub in the velocity of the plane.
Vp = 600 / (Tcar - 5.5)

(600 / (600 / (Tcar - 5.5))) + 5.5 ) = 1 / Tcar

Now.. Ive made a mess of the algebra, because its late... and im tired. you can use the solver in a TI calucaltor, or do a better job of setting up equations, but what youre left with that this poing, is one equation, and one unknown, and once you find that, you can go back to the formular for the averages, and sub in Tcar to find the answer.

Good Luck !