Question

rob is investigating the effects of font size on the number of words that fit on a page. he changes the font size on an essay and records the number of words on one page of the essay. the table shows his data. words per page font size 14 12 16 10 12 14 16 18 24 22 word count 352 461 340 407 435 381 280 201 138 114 which equation represents the approximate line of best fit for data, where x represents font size and y represents the number of words on one page? y = -55x + 407 y = -41x + 814 y = -38x + 922 y = -26x + 723

Explanation:

Step1: Calculate the slope

We can use two points to estimate the slope. Let's take \((x_1,y_1)=(10,407)\) and \((x_2,y_2)=(24,138)\)
The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\)
\(m=\frac{138 - 407}{24 - 10}=\frac{-269}{14}\approx - 19.21\) (This is a rough check, another way is to use the general trend)
Another approach: assume the linear equation \(y = mx + b\). We can also use the fact that when \(x = 10,y = 407\); when \(x=24,y = 138\)
Substitute into \(y=mx + b\):
\(

$$\begin{cases}407 = 10m + b\\138=24m + b\end{cases}$$

\)
Subtract the first equation from the second: \((24m + b)-(10m + b)=138 - 407\)
\(24m + b-10m - b=-269\)
\(14m=-269\), \(m\approx - 19.21\) (not the most accurate, but we can check the options)
Let's check each option by substituting \(x\) values.
For option \(y=-41x + 814\)
When \(x = 10\), \(y=-41\times10 + 814=404\) (close to \(407\))
When \(x = 12\), \(y=-41\times12 + 814=-492+814 = 322\) (not close to \(461\) and \(435\))
For option \(y=-38x + 922\)
When \(x = 10\), \(y=-38\times10+922 = 542\) (not close)
When \(x = 12\), \(y=-38\times12 + 922=-456+922 = 466\) (close to \(461\) and \(435\))
When \(x = 14\), \(y=-38\times14+922=-532 + 922=390\) (close to \(352\) and \(381\))
When \(x = 16\), \(y=-38\times16+922=-608+922 = 314\) (close to \(340\) and \(280\))
When \(x = 18\), \(y=-38\times18+922=-684+922 = 238\) (close to \(201\))
When \(x = 22\), \(y=-38\times22+922=-836+922 = 86\) (close to \(114\))
When \(x = 24\), \(y=-38\times24+922=-912+922 = 10\) (not so close, but overall better fit than other options)

Answer:

\(y=-38x + 922\)