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Answer :
We begin with the given data points for the number of months since the start of the build ([tex]$x$[/tex]) and the percentage of the house left to build ([tex]$y$[/tex]):
[tex]$$
\begin{array}{cc}
x & y \\
\hline
0 & 100 \\
1 & 86 \\
2 & 65 \\
3 & 59 \\
4 & 41 \\
5 & 34 \\
\end{array}
$$[/tex]
Since a linear function of the form
[tex]$$
y = mx + b
$$[/tex]
is to be used to model this relationship, we need to determine the slope ([tex]$m$[/tex]) and the [tex]$y$[/tex]-intercept ([tex]$b$[/tex]).
### Step 1. Calculate the Slope
The slope is given by the formula
[tex]$$
m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2},
$$[/tex]
where [tex]$n$[/tex] is the number of data points.
1. There are [tex]$n = 6$[/tex] data points.
2. The sums needed are:
- [tex]$\sum x = 0 + 1 + 2 + 3 + 4 + 5$[/tex]
- [tex]$\sum y = 100 + 86 + 65 + 59 + 41 + 34$[/tex]
- [tex]$\sum (xy)$[/tex] is the sum of the product of each pair [tex]$(x,y)$[/tex].
- [tex]$\sum (x^2)$[/tex] is the sum of the squares of [tex]$x$[/tex].
By substituting and calculating, we determine that the slope is approximately
[tex]$$
m \approx -13.457.
$$[/tex]
### Step 2. Calculate the [tex]$y$[/tex]-intercept
The [tex]$y$[/tex]-intercept is given by
[tex]$$
b = \frac{\sum y - m\sum x}{n}.
$$[/tex]
Using the previously calculated slope and appropriate sums, we find that
[tex]$$
b \approx 97.810.
$$[/tex]
### Step 3. Compare with the Given Options
The candidate functions given are:
1. [tex]$\displaystyle y = -13.5x + 97.8$[/tex]
2. [tex]$\displaystyle y = -13.5x + 7.3$[/tex]
3. [tex]$\displaystyle y = 97.8x - 13.5$[/tex]
4. [tex]$\displaystyle y = 7.3x - 97.8$[/tex]
Our calculated slope of approximately [tex]$-13.457$[/tex] and [tex]$y$[/tex]-intercept of approximately [tex]$97.810$[/tex] closely match the values in option 1, which is
[tex]$$
y = -13.5x + 97.8.
$$[/tex]
### Conclusion
Since the computed values are nearly identical to those in the first candidate function, the linear function that best models the data is:
[tex]$$
\boxed{y=-13.5x+97.8.}
$$[/tex]
[tex]$$
\begin{array}{cc}
x & y \\
\hline
0 & 100 \\
1 & 86 \\
2 & 65 \\
3 & 59 \\
4 & 41 \\
5 & 34 \\
\end{array}
$$[/tex]
Since a linear function of the form
[tex]$$
y = mx + b
$$[/tex]
is to be used to model this relationship, we need to determine the slope ([tex]$m$[/tex]) and the [tex]$y$[/tex]-intercept ([tex]$b$[/tex]).
### Step 1. Calculate the Slope
The slope is given by the formula
[tex]$$
m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2},
$$[/tex]
where [tex]$n$[/tex] is the number of data points.
1. There are [tex]$n = 6$[/tex] data points.
2. The sums needed are:
- [tex]$\sum x = 0 + 1 + 2 + 3 + 4 + 5$[/tex]
- [tex]$\sum y = 100 + 86 + 65 + 59 + 41 + 34$[/tex]
- [tex]$\sum (xy)$[/tex] is the sum of the product of each pair [tex]$(x,y)$[/tex].
- [tex]$\sum (x^2)$[/tex] is the sum of the squares of [tex]$x$[/tex].
By substituting and calculating, we determine that the slope is approximately
[tex]$$
m \approx -13.457.
$$[/tex]
### Step 2. Calculate the [tex]$y$[/tex]-intercept
The [tex]$y$[/tex]-intercept is given by
[tex]$$
b = \frac{\sum y - m\sum x}{n}.
$$[/tex]
Using the previously calculated slope and appropriate sums, we find that
[tex]$$
b \approx 97.810.
$$[/tex]
### Step 3. Compare with the Given Options
The candidate functions given are:
1. [tex]$\displaystyle y = -13.5x + 97.8$[/tex]
2. [tex]$\displaystyle y = -13.5x + 7.3$[/tex]
3. [tex]$\displaystyle y = 97.8x - 13.5$[/tex]
4. [tex]$\displaystyle y = 7.3x - 97.8$[/tex]
Our calculated slope of approximately [tex]$-13.457$[/tex] and [tex]$y$[/tex]-intercept of approximately [tex]$97.810$[/tex] closely match the values in option 1, which is
[tex]$$
y = -13.5x + 97.8.
$$[/tex]
### Conclusion
Since the computed values are nearly identical to those in the first candidate function, the linear function that best models the data is:
[tex]$$
\boxed{y=-13.5x+97.8.}
$$[/tex]
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