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: Analyze linear models

First, check the linear models \( y = 26.9x - 1.3 \) and \( y = 27x - 4 \). Let's plug in \( x = 2 \):
For \( y = 26.9x - 1.3 \), \( y = 26.9(2)-1.3 = 53.8 - 1.3 = 52.5 \). The actual \( y \) is 50.
For \( y = 27x - 4 \), \( y = 27(2)-4 = 54 - 4 = 50 \). This matches \( x = 2 \).
Check \( x = 4 \):
For \( y = 27x - 4 \), \( y = 27(4)-4 = 108 - 4 = 104 \). The actual \( y \) is 110. Close.
Check \( x = 6 \):
\( y = 27(6)-4 = 162 - 4 = 158 \). Actual \( y = 160 \). Close.
Check \( x = 8 \):
\( y = 27(8)-4 = 216 - 4 = 212 \). Actual \( y = 210 \). Close.
Check \( x = 10 \):
\( y = 27(10)-4 = 270 - 4 = 266 \). Actual \( y = 270 \). Close.
Check \( x = 12 \):
\( y = 27(12)-4 = 324 - 4 = 320 \). Actual \( y = 320 \). Perfect match.

Step2: Analyze exponential models

Check exponential models \( y = 47(1.191)^x \) and \( y = 34(1.204)^x \). For \( x = 2 \):
\( y = 47(1.191)^2 \approx 47(1.418) \approx 66.65 \), which is more than 50.
\( y = 34(1.204)^2 \approx 34(1.449) \approx 49.27 \), close but not 50. The linear model \( y = 27x - 4 \) matches better across all \( x \).

Answer:

\( y = 27x - 4 \)