Question

**WILL GIVE BRANLIEST** 15 points
In the first week of training Alana ran an average of 12 minutes per mile. Later, in week five of training she ran an average of 6 minutes per mile.
In the first week of training Kelsa ran an average of 10 minutes per mile. Later, in week eleven of training she ran an average of 5 minutes per mile.
Assuming that Alana and Kelsa continue to train and improve their times at the same rate your task is to determine which week they will have the same average minutes per mile. We will assume that the relationship is linear as they will be training for a maximum of 25 weeks. To complete this task follow the steps below.

1. Determine the equation of a line in standard form that represents Alana’s training progress. Her progress corresponds to the points (1, 12) and (5,6).

2. Determine the equation of a line in standard form that represents Kelsa’s training progress. Her progress corresponds to the points (1, 10) and (11, 5)

3. Solve the system of equations (you must show all your work to receive full credit).

4. In which week will Alana and Kelsa have the same average minutes per mile?

5. If Alana and Kelsa continue to train until week 25, what will their times be?

6. Do you believe a linear model best represents the relationship of the time of the runners and the weeks that passed?(Hint: look at question 5). What do you think this says about problems in the real world? Justify your thoughts in 3-4 sentences.

Answer 1

Question 1

First we compute the gradient of the line as follows;

$m=\frac{y_2-y_1}{x_2-x_1}= \frac{6-12}{5-1} =-\frac{6}{2} =-\frac{3}{2}.$

Next step is to use the  general gradient intercept-form of the equation of the straight line using the gradient and the point(1,12) to find the equation of the line.

$y=mx+c\\=>12=-\frac{3}{2}(1)+ c \\=>\frac{24}{2}+ \frac{3}{2}=\frac{27}{2} c\\ =>y=-\frac{3}{2}x+ \frac{27}{2}$

The next step is to express the line in standard form. This will be achieved by multiplying the gradient intercept form by 2.

In standard form the equation of the line representing Alana's training is;

$2y=-3x+27$

Question 2

First we compute the gradient of the line as follows;

$m=\frac{y_2-y_1}{x_2-x_1}= \frac{5-10}{11-1} =-\frac{5}{10} =-\frac{1}{2}.$

Next step is to use the  general gradient intercept-form of the equation of the straight line using the gradient and the point(1,12) to find the equation of the line.

$y=mx+c\\=>10=-\frac{1}{2}(1)+ c \\=>\frac{20}{2}+ \frac{1}{2}=\frac{21}{2} c\\ =>y=-\frac{1}{2}x+ \frac{21}{2}$

The next step is to express the line in standard form. This will be achieved by multiplying the gradient intercept form by 2.

In standard form the equation of the line representing Alana's training is;

$2y=-x+21.)$

Question 3

We will solve the two equations simultaneously by first eliminating y and finding the x-coordinate which represents the time.

$2y=-3x+27....(1)\\2y=-x+21-....(2)\\(1)-(2)\\=>2y-2y=-3x-(-x)+27-21\\=>0=-2x+6\\=>2x=6\\x=3$

We substitute the value of the time coordinate  in equation (1) to get the value for y.

$2y=-3x+27\\=>2y=-3(3)+27\\=>2y=18\\=>y=9$.

The solution to these equations is 3 minutes per mile at 9 weeks,i.e (9,3)

Question 4

Alana and Kelsey have the same average minutes per mile in week 9.

Question 5

At 25 weeks, Alana's time will be found by using an x value of 25 in he equation representing her progress;

$2y=-3x+27\\=>2y=-3(25)+27\\2y=-75+27=-24\\y=-24.$

At 25 weeks, Kelsey's time will be found by using an x value of 25 in he equation representing her progress;

$2y=-x+21\\=>2y=-(25)+21\\2y=-4\\y=-2.$

Question 6

The linear models are not best representations of the relationship of the runners average times and the week that pass. Alana's model shows that at 25 weeks she has average minutes of -24 minutes per mile. Kelsey's model shows that at 25 weeks she has average minutes of -4 minutes per mile. Negative time makes no physical sense.