A plane flies 720mi against a steady 30-mi/h headwind

[dkarolasz]

3. A plane flies 720mi against a steady 30-mi/h headwind and then returns to the same point with the wind. If the entire trip takes 10 h, what is the plane's speed in still air?

Let r = rate of plane in still air.

Flying AGAINST the wind, the rate is r - 30 (because the headwind decreases the plane's speed). It will take 720 / (r - 30) hours to fly 720 miles.

Flying WITH the wind, the rate is r + 30 (because the tailwind increases the plane's speed). It will take 720 / (r + 30) hours to fly 720 miles.

The total time for the round trip is 10 hours. So,

[720 / (r - 30)] + [720 / (r + 30)] = 10

Solve for r.

I'm having a hard time with it

[720 / (r - 30)] + [720 / (r + 30)] = 10

Solve for r.

I did 720 = r-30
+30 +30
_______________
750 = r

720 = r+30
-30 -30
___________

690 = r

I'm not getting it..

 

[soroban]

Re: Help!!

Hello, dkarolasz!

Your set-up is excellent!
Thank you for showing us your reasoning and your work.

3. A plane flies 720mi against a steady 30-mi/h headwind
and then returns to the same point with the wind.
If the entire trip takes 10 h, what is the plane's speed in still air?

Let r = rate of plane in still air.

Flying AGAINST the wind, the rate is \(\displaystyle r\,-\,30\)
(because the headwind decreases the plane's speed).
. . It will take: \(\displaystyle \,\frac{720}{r\,-\,30}\) hours to fly 720 miles.

Flying WITH the wind, the rate is \(\displaystyle r\,+\,30\)
(because the tailwind increases the plane's speed).
. . It will take: \(\displaystyle \,\frac{720}{r\,+\,30}\) hours to fly 720 miles.

The total time for the round trip is 10 hours.
. . So: \(\displaystyle \L\:\frac{720}{r\,-\,30} \,+\,\frac{720}{r\,+\,30} \;=\;10\)

Solve for r. . Good!

Multiply through by the common denominator: \(\displaystyle (r\,-\,30)(r\,+\,30)\)

. . \(\displaystyle \L(r\,-\,30)(r\,+\,30)\cdot\frac{720}{r\,-\,30} \,+\,(r\,-\,30)(r\,+\,30)\cdot\frac{720}{r\,+\,30} \;= \;(r\,-\,3)(r\,+\,30)\cdot720\)

This simplifies to: \(\displaystyle \:10r^2\,-\,1440r\,-\,9000\;=\;0\)

Divide by 10: \(\displaystyle \:r^2\,-\,144r\,-\,900\;=\;0\)

. . which factors: \(\displaystyle \r\,-\,150)(r\,+\,6)\;=\;0\)

. . and has roots: \(\displaystyle \:r\:=\:150,\:-6\)


Therefore, the plane's ground speed is 150 mph

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Check

Flying against the wind, its speed was: \(\displaystyle \,150\,-30\:=\:120\text{ mph.}\)
. . To fly 720 miles, it took: \(\displaystyle \frac{720}{120}\:=\:6\text{ hours.}\)

Flying with the wind, its speed was: \(\displaystyle \,150\,+\,30\:=\:180\text{ mph.}\)
. . To fly 720 miles, it took: \(\displaystyle \frac{720}{180} \:=\:4\text{ hours.}\)

The entire flight took: \(\displaystyle \,6\,+\,4\:=\:10\text{ hours}\) . . . check!

 

[Mrspi]

Re: Help!!

Hello, dkarolasz!

Your set-up is excellent!
Thank you for showing us your reasoning and your work.

Thanks, Soroban....that is MY setup copied from a previous response to this student.

 

[dkarolasz]

yes that was Mrspi work, I copied and pasted it to a new topic, because no one was going back and helping me with it. I couldn't finger it out. Thanks to you both I understand it quite well now.

Thanks again...

 

[stapel]

...no one was going back and helping me with it.

I think you mean that people were "only" trying to help you learn, giving you the set-up, reasoning, instructions, and hints. But the tutors were waiting for you to stop making random guesses and invest some effort into your own learning. Nobody was doing things for you.

I understand it quite well now.

It's interesting how completely "at sea" you are when you are "only" given step-by-step instructions, explanations, and set-ups, but how you are able to completely and correctly solve things "on your own" once sorobon gives you the complete correct worked solution. :shock:

FYI: This is why legitimate tutors don't generally provide complete worked solutions: students almost never learn from them. If we are wrong in your case, then we will of course shortly begin seeing something from you other than apparently random guesses and reposts of other peoples' work disingenuously presented as your own; indeed, your posts will start showing some learning and progress on your part, for which we will all be very happy.

Thank you.

Eliz.