'uniform rate' prob: How long did he take on each lap?

santasad13

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Jan 26, 2006
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here is another one of those chart problems that stump me and my parents

Jamie ran 2 laps around a track in 99 seconds. How long did it take him to run each lap if he ran the first lap at 8.5 meters per second ("m/s") and the second lap at 8 m/s.

the formula is rate X time=distance.
 

stapel

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Set up the same chart as last time. The two rows will be "first lap" and "second lap". Fill in the known rates. Plug in "d" for the (unknown) distance.

Note that, if d = rt, then t = d/r. Use this to form expressions, in terms of the variable "d", for each of the two times "t". Plug these into the chart.

You know that the sum of the two times is 99, so create the equation. Solve for the distance "d". Back-solve for the times of each lap.

If you get stuck, please reply showing how far you have gotten. Thank you.

Eliz.
 

tkhunny

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Quit hating 8th grade. Hate is no good. Spokane is a GREAT place. Enjoy it.

Write stuff down and ponder it. Do NOT try to figure out, memorize, or internalize some arbitrary structure. Find a way of thinking that works logically and consistently FOR YOU. You can worry about fitting it into some arbitrary structure AFTER you understand what you are doing.

Distance = Rate * Time

"Jamie ran 2 laps around a track in 99 seconds."

(Two Laps) = (Average Rate)*(99 seconds)

"he ran the first lap at 8.5 m/s"

(First Lap) = (8.5 m/s)*(Some Seconds)

"he ran the second lap at 8 m/s"

(Second Lap) = (8.0 m/s)*([99-Some] Seconds)

The ONLY thing close to tricky in there was noting that the first lap took SOME time and the second lap took the rest of the time.

Now we stop and think. Are there more clues? No. Okay, then we ahve to come up with something...OH, as I was typing, it dawned on me that these are LAPS! Generally, one lap is not longer or shorter than another lap. They are the same length!!! We can use that.

First Lap = Second LAP

(First Lap) = (8.5 m/s)*(Some Seconds)
(Second Lap) = (8.0 m/s)*([99-Some] Seconds)

So, ...

(8.5 m/s)*(Some Seconds) = (8.0 m/s)*([99-Some] Seconds)

Now THAT look slike somethign that can be solved.

(8.5)*(Some Seconds) = (8.0)*([99-Some] Seconds)

(8.5)*(Some) = (8.0)*([99-Some])

(8.5)*(Some) = (8.0)*99 - (8.0)*Some

(8.5)*(Some) + (8.0)*Some = (8.0)*99

16.5*Some = 792

Some = 792/16.5 = 48 = The seconds required for the fist lap.

99 - Some = 99 - 48 = 51 = The seconds required for the second lap

Notice that 51 > 48. This is good, since the second lap was at a lower rate.

Just think it through. Make it make sense in your head. Write stuff down and let the notation help you.
 

soroban

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Jan 28, 2005
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Hello, santasad13!

Here is another one of those chart problems that stump me and my parents.

Jamie ran 2 laps around a track in 99 seconds.
How long did it take him to run each lap
if he ran the first lap at 8.5 m/s and the second lap at 8 m/s.

The formula is: Rate x Time = Distance

We know the rates (speeds) for the first and second laps.
Code:
        | Rate x Time  =  Dist. |
- - - - + - - -+ - - -| - - - - +
1st lap | 8.5  |      |         |
- - - - + - - -+ - - -+ - - - - +
2nd lap |  8   |      |         |
- - - - + - - -+ - - -+ - - - - +

Let \(\displaystyle x\\) = time to run the first lap (in seconds).
Then \(\displaystyle 99\,-\,x\) = time to run the second lap.
Code:
        | Rate x Time  =  Dist. |
- - - - + - - -+ - - -| - - - - +
1st lap | 8.5  |   x  |         |
- - - - + - - -+ - - -+ - - - - +
2nd lap |  8   | 99-x |         |
- - - - + - - -+ - - -+ - - - - +

Using the formula, we fill in the last column.
Code:
        | Rate x Time  =  Dist. |
- - - - + - - -+ - - -| - - - - +
1st lap | 8.5  |   x  |   8.5x  |
- - - - + - - -+ - - -+ - - - - +
2nd lap |  8   | 99-x | 8(99-x) |
- - - - + - - -+ - - -+ - - - - +

Since the distances for the two laps are equal,

. . our equation is: \(\displaystyle \L\:8.5x\;=\;8(99\,-\,x)\)

Can you finish it now?

 

santasad13

Junior Member
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Jan 26, 2006
Messages
50
Thank you Eliz and tkhunny-oh, and soroban.and it is hard not to :lol: by the way, have you ever been to spokane? just curious.
 

tkhunny

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Spokane? Sure. I lived in five different places and worked out in the Valley and Downtown. Just for the record, NO, I wasn't hiding from landlords, we just kept moving because it was fun! Growing families and better jobs will do that. :D Spokane is where I got my first MATH job. :wink:

What part of town are you from? (Please don't be too specific!)

South Hill?
Valley?
Off the hill by SFCC?
Indian Trail?
Country Homes?
Colville?
Hilliard?
Way out toward Moses Lake?

They have decent schools all over the place. I know only one person who was expelled from all the local schools and failed to graduate.
 

santasad13

Junior Member
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Jan 26, 2006
Messages
50
i live in the south hill area. and i dont go to a public school, but a private school
 

santasad13

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Jan 26, 2006
Messages
50
did his name happen to be J__ W______?
 

tkhunny

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That was a little too specific. Don't do that.

I'm just rrying to tell you that you have opportunities to succeed. You are in a fine community and likely have a fine teacher and adequate textbooks. No need for hating.

Let's learn some mathematics!
 
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