Question

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

Explanation:

Step1: Check linear models

For a linear model \(y = mx + b\), the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
Take \(x_1 = 2,y_1 = 50\) and \(x_2=4,y_2 = 110\). Then \(m=\frac{110 - 50}{4 - 2}=\frac{60}{2}=30\).
For \(y = 26.9x-1.3\), when \(x = 2\), \(y=26.9\times2-1.3=53.8 - 1.3 = 52.5
eq50\).
For \(y = 27x-4\), when \(x = 2\), \(y=27\times2-4=54 - 4=50\). When \(x = 4\), \(y=27\times4-4=108 - 4 = 104
eq110\).

Step2: Check exponential models \(y = a\cdot b^x\)

For \(y = 47(1.191)^x\), when \(x = 2\), \(y=47\times(1.191)^2=47\times1.418481\approx66.67
eq50\).
For \(y = 34(1.204)^x\), when \(x = 2\), \(y = 34\times(1.204)^2=34\times1.449616\approx49.29\approx50\). When \(x=4\), \(y=34\times(1.204)^4\). \((1.204)^2\approx1.4496\), \((1.204)^4=(1.204)^2\times(1.204)^2\approx1.4496\times1.4496\approx2.10\), \(y = 34\times2.10 = 71.4
eq110\). But if we calculate more accurately: \((1.204)^4=1.204\times1.204\times1.204\times1.204=(1.204^2)^2=(1.449616)^2\approx2.101\), \(y=34\times2.101 = 71.434
eq110\). Wait, actually, if we consider the general trend.
Let's use another approach. Assume \(x = 2,y = 50\); \(x=4,y = 110\); \(x = 6,y = 160\).
The ratio for exponential \(r=\frac{y_2}{y_1}\) (for \(x_2=x_1 + 2\)). \(\frac{110}{50}=2.2\), \(\frac{160}{110}\approx1.45\) (not a constant ratio for exponential). But if we check the differences: \(110 - 50=60\), \(160 - 110 = 50\), \(210-160 = 50\), \(270 - 210=60\), \(320 - 270 = 50\). Wait, no, actually, if we calculate the slope more accurately for linear.
Take \(x_1=2,y_1 = 50\); \(x_2 = 12,y_2=320\). Slope \(m=\frac{320 - 50}{12 - 2}=\frac{270}{10}=27\). Then \(y-50=27(x - 2)\), \(y=27x-54 + 50=27x - 4\). When \(x = 4\), \(y=27\times4-4=108 - 4=104\) (there is some error due to data approximation, but compared to other models)

Answer:

\(y = 27x-4\)