Part A:
is the equation of the line.
Part B: Anika worked 17.5 hours on day 0.
Part C: The setup time decreases with 1 hour and 15 min per day.
Explanation:
Part A: Let us find the equation of the line using the points (2,15) and (6,10)
The equation of the line is given by,
![y-y_{1}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \cdot\left(x-x_{1}\right)](https://tex.z-dn.net/?f=y-y_%7B1%7D%3D%5Cfrac%7By_%7B2%7D-y_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D%20%5Ccdot%5Cleft%28x-x_%7B1%7D%5Cright%29)
Substituting, we get,
![y-15=\frac{10-15}{6-2} (x-2)](https://tex.z-dn.net/?f=y-15%3D%5Cfrac%7B10-15%7D%7B6-2%7D%20%28x-2%29)
Simplifying, we have,
![y-15=-\frac{5}{4} (x-2)](https://tex.z-dn.net/?f=y-15%3D-%5Cfrac%7B5%7D%7B4%7D%20%28x-2%29)
![y-15=-\frac{5}{4} x+\frac{5}{2}](https://tex.z-dn.net/?f=y-15%3D-%5Cfrac%7B5%7D%7B4%7D%20x%2B%5Cfrac%7B5%7D%7B2%7D)
Subtracting both sides by 15, we get,
![y=-\frac{5}{4} x+\frac{35}{2}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B5%7D%7B4%7D%20x%2B%5Cfrac%7B35%7D%7B2%7D)
Hence, the equation of the line is ![y=-\frac{5}{4} x+\frac{35}{2}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B5%7D%7B4%7D%20x%2B%5Cfrac%7B35%7D%7B2%7D)
Part B: To determine the number of hours Anika worked on day 0, let us substitute x = 0 in the equation of the line.
Thus, we get,
![y=\frac{35}{2}\\y=17.5](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B35%7D%7B2%7D%5C%5Cy%3D17.5)
Hence, Anika worked 17.5 hours on day 0.
Part C: The setup time decrease per day can be determined using the slope.
The formula for slope is given by
![m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7By_%7B2%7D-y_%7B1%7D%7D%7Bx_%7B2%7D-x_%7B1%7D%7D)
Substituting , we have,
![m=\frac{10-15}{6-2}=-\frac{5}{4}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B10-15%7D%7B6-2%7D%3D-%5Cfrac%7B5%7D%7B4%7D)
Converting the fraction into hours and minutes, we get,
![-\frac{5}{4}\times60=-75 min](https://tex.z-dn.net/?f=-%5Cfrac%7B5%7D%7B4%7D%5Ctimes60%3D-75%20min)
Since, 1 hour = 60 min
Subtracting 60 and 75 = 15 min
Thus, the setup time decreases with 1 hour and 15 min per day.