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A construction manager is monitoring the progress of building a new house. The scatterplot and table below show the number of months since the start of the build and the percentage of the house still left to build. A linear function can be used to model this relationship.

[tex]
\[
\begin{tabular}{|c|c|}
\hline
\begin{tabular}{c}
Number of \\
Months Since \\
Start of Build, $x$
\end{tabular}
&
\begin{tabular}{c}
Percentage of \\
House Left \\
to Build, $y$
\end{tabular} \\
\hline
0 & 100 \\
\hline
1 & 86 \\
\hline
2 & 65 \\
\hline
3 & 59 \\
\hline
4 & 41 \\
\hline
5 & 34 \\
\hline
\end{tabular}
\]
[/tex]

Which function best models the data?

A. [tex]y = -13.5x + 97.8[/tex]

B. [tex]y = -13.5x + 7.3[/tex]

C. [tex]y = 97.8x - 13.5[/tex]

D. [tex]y = 7.3x - 97.8[/tex]

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]

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Rewritten by : Barada