Question

the table shows the estimated number of lines of code written by computer programmers per hour when x people are working. productivity
people working lines of code written hourly
2 50
4 110
6 160
8 210
10 270
12 320
which model best represents the data?
y = 47(1.191)^x
y = 34(1.204)^x
y = 26.9x - 1.3
y = 27x - 4

Explanation:

Step1: Check linear - form

We first check if the data is linear. For a linear model $y = mx + b$, the slope $m=\frac{\Delta y}{\Delta x}$.
For $x$ values $2$ and $4$: $\Delta x=4 - 2=2$, $y$ values are $50$ and $110$, $\Delta y=110 - 50 = 60$, slope $m_1=\frac{60}{2}=30$.
For $x$ values $4$ and $6$: $\Delta x=6 - 4 = 2$, $y$ values are $110$ and $160$, $\Delta y=160 - 110=50$, slope $m_2=\frac{50}{2}=25$. Since the slopes are not exactly the same but close, we test the linear models.
For $y = 26.9x-1.3$, when $x = 2$, $y=26.9\times2-1.3=53.8 - 1.3=52.5$. When $x = 4$, $y=26.9\times4-1.3=107.6-1.3 = 106.3$.
For $y = 27x-4$, when $x = 2$, $y=27\times2-4=54 - 4=50$. When $x = 4$, $y=27\times4-4=108 - 4 = 104$.

Step2: Check exponential - form

For an exponential model $y = a(b)^x$. For $y = 47(1.191)^x$, when $x = 2$, $y=47\times(1.191)^2=47\times1.418481\approx66.67$.
For $y = 34(1.204)^x$, when $x = 2$, $y=34\times(1.204)^2=34\times1.449616\approx49.29$.

Step3: Compare with data

By substituting more $x$ - values from the table into the four models, we find that the linear model $y = 27x - 4$ gives values that are closest to the values in the table.

Answer:

$y = 27x - 4$