Answer: Alana's training experience in (weeks, minutes per mile) is (1, 12) and (5, 6).
Kelsa's training experience is (1, 10), (11, 5).
Find:
A linear equation in standard form that describes Alana's experience
A linear equation in standard form that describes Kelsa's experience
A solution to the pair of linear equations
The week in which the number of minutes per mile is the same
Minutes per mile for Alana and Kelsa at week 25
Comment on the use of a linear model in this context
Solution:
Please note that in parts 1 and 2 a "standard form" linear equation has a positive leading coefficient and the numbers have no common factor.
1. A linear equation can be written from two points (x1, y1) and (x2, y2) as ...
... (y2 -y1)(x -x1) -(x2 -x1)(y -y1) = 0
Using the given points in order, this becomes
... (6-12)(x -1) -(5 -1)(y -12) = 0 . . . . above equation with values filled in
... -6x +6 -4y +48 = 0 . . . . . . . . . . eliminate parentheses
... 3x +2y -27 = 0 . . . . . . . . . . . . . divide by -2 to make x-coefficient positive
... 3x +2y = 27 . . . . . . . . . . . . . . . . Alana's training experience
2. In similar fashion, we can write the equation for Kelsa's experience.
... (5 -10)(x -1) -(11 -1)(y -10) = 0
... -5x +5 -10y +100 = 0 . . . . . . eliminate parentheses
... x +2y -21 = 0 . . . . . . . . . . . . . divide by -5
... x +2y = 21 . . . . . . . . . . . . . . . . Kelsa's training experience
3. We note that both equations have the term 2y, so we can subtract the second equation from the first to eliminate that term:
... (3x +2y) -(x +2y) = (27) -(21)
... 2x = 6 . . . . . . . simplify
... x = 3 . . . . . . . . divide by 2
... 3·3 +2y = 27 . . . substitute x=3 into Alana's equation
... 2y = 18 . . . . . . . . subtract 9
... y = 9 . . . . . . . . . . divide by 2
The solution is (weeks, minutes) = (x, y) = (3, 9).
4. The solution of part 3 tells us both have an average time of 9 minutes per mile in week 3 of training.
5. Substituting x=25 into the equations of parts 1 and 2, we get ...
... 3·25 +2y = 27
... y = (27 -75)/2 = -24 . . . . Alana's time per mile at week 25
... 25 + 2y = 21
... y = (21 -25)/2 = -2 . . . . . Kelsa's time per mile at week 25
6. A linear model makes no sense in this context. At some point a linear equation for minutes per mile will always have a value of 0. We know that marathon runners will never have a time of zero minutes per mile. Under the best of conditions, we expect training to improve time toward some asymptotic limit determined by body mechanics. A linear model is completely inappropriate for describing such training effects. Not every real-world problem is appropriately represented by a linear equation.
Step-by-step explanation:
