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Scott, MIT Graduate

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A. The record for the fastest mile run by a person has

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A. The record for the fastest mile run by a person has been decreasing over time. Back around 1850, when record times were first recorded, the fastest mile time recorded was about 5 minutes. By around 1950, the fastest mile time recorded was about 4 minutes.

Note: Assume that the fastest mile time (run by a person) has been decreasing linearly over time since 1850.

1. Find the function R(t) that models the (assumed) linear decrease in the fastest timed mile records over time t, where t = 0 corresponds to 1850.

2. Use your function R(t) to predict the following:

a. The record for the fastest mile run in the year 2000

b. The year in which a mile will be run in 3 minutes

c. The year in which a mile will be run in 2 minutes

d. The year in which a mile will be run in 1 minute

e. The year in which a mile will be run in 0 minutes

3. Explain the limitations of your model.

Note to teacher: I do know that a mathematical model can be developed.

OK, this is not algebra so much as it is just making a good old fashioned line graph and predicting where the line will cross 3 mins and 2 mins. You can do it by hand or you can do it by enterring the information into two columns of excel, one column being the year of the resord speed and the other being the time, and then telling it to make a graph for you and seeing where it throws the line outpast 3 minutes and 2 minutes.

If I have the dates and the times, I can probably set it up, but if you wish to try it yourself first, please do, because you will need to be able to do this yourself eventually. I'll opt out for now but if you respond you want help and can give me the times, I will step back in.

This problem comes from College Algebra and the thing that I have to accomplish is to create an algebraicl model that consists of a linear equation to arrive at each answer. The two things the problem give are: 1) 1850 is the start date for recording time, and 2) Mile time has been decreasing in a linear fashion since 1850. For example, between 1850 and 1950, one minute was shaved off the fastest mile run. Following along those lines, the model could show 1/2 minute is shaved off of the run time by 2000, etc. The model could perhaps include 'x' (to represent the number of years).

I'll get this moved to the math category right away. This is not accurately solved with line graphs. Algebra is much more accurate.

Back around 1850, when record times were first recorded, the fastest mile time recorded was about 5 minutes. By around 1950, the fastest mile time recorded was about 4 minutes. Note: Assume that the fastest mile time (run by a person) has been decreasing linearly over time since 1850.

1. Find the function R(t) that models the (assumed) linear decrease in the fastest timed mile records over time t, where t = 0 corresponds to 1850.

slope = rise/run

= (4-5)/(1950-1850)

= -1/100

using slope intercept form:

R(t) = -1/100 t + 5

2. Use your function R(t) to predict the following:

a. The record for the fastest mile run in the year 2000

Plug in t = 150, since it's 150 years after 1850:

R(150) = -1/100 * 150 + 5

= 3.5 minutes

b. The year in which a mile will be run in 3 minutes

Set R = 3:

3 = -1/100 t + 5

Subtract 5:

-2 = -1/100 t

Multiply by -100:

t = 200

That's year 2050

c. The year in which a mile will be run in 2 minutes

Set R = 2:

2 = -1/100 t + 5

Subtract 5:

-3 = -1/100 t

Multiply by -100:

t = 300

That's year 2150

d. The year in which a mile will be run in 1 minute

Set R = 1:

1 = -1/100 t + 5

Subtract 5:

-4 = -1/100 t

Multiply by -100:

t = 400

That's year 2250

e. The year in which a mile will be run in 0 minutes

Set R = 0:

0 = -1/100 t + 5

Subtract 5:

-5 = -1/100 t

Multiply by -100:

t = 500

That's year 2350

3. Explain the limitations of your model.

Obviously, the time cannot be 0 minutes. The actual running time cannot be simulated with a simple linear equation, since there is probably a limit on how low the time can go. The linear function might be good for the first 100 or 200 years, but after that it won't work well.

Let me know if you have any questions. If you're set, thanks for accepting. Positive feedback is appreciated :)

Well, what an excellent demonstration, however, before I accept, can you explain why slope-intercept form of an equation would be appropriate for this type of problem. I ran through your examples, and the choice that you used looks like it can very well predict future outcomes which is what we have to show. Thanks!

P>S> Sorry about the misspelled words in my previous question.

I used slope intercept form because we are given the time at year t = 0. The time value at t = 0 is the y intercept, by definition. You could use point slope form and solve it, but you'd get the same answer:

y-5 = -1/100(x-0)

y-5 = -1/100 x

y = -1/100 x + 5

Let me know if you have any other questions about this before you accept. Thanks!