Question

rah is investigating the effects of font size on the number of words that fit on a page. he changes the font size on his essay and records the number of words on one page of the essay. the table shows his data.
words per page
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 and y - intercept approximately

We can use two points to estimate. Let's take (10,407) and (24,138).
The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
\(m=\frac{138 - 407}{24 - 10}=\frac{-269}{14}\approx - 19.2\).
Another way is to use the formula \(y=mx + b\). Substitute \(x = 10,y = 407\) into \(y=mx + b\), \(407=10m + b\). Substitute \(x = 24,y = 138\) into \(y=mx + b\), \(138=24m + b\).
Subtract the first equation from the second: \((138 - 407)=(24m + b)-(10m + b)\), \(-269 = 14m\), \(m\approx - 19.2\). If we assume we use a more accurate method (such as linear regression calculator or software), for the given data.
Let's check each option:
For \(y=-38x + 922\):
When \(x = 10\), \(y=-38\times10 + 922=922-380 = 542\) (not close to 407).
For \(y=-26x + 723\):
When \(x = 10\), \(y=-26\times10+723=723 - 260=463\) (not close to 407).
For \(y=-55x + 407\):
When \(x = 12\), \(y=-55\times12 + 407=-660 + 407=-253\) (not valid as word - count is positive).
For \(y=-41x + 814\):
When \(x = 10\), \(y=-41\times10+814=814 - 410 = 404\approx407\). When \(x = 12\), \(y=-41\times12 + 814=-492+814 = 322\). When \(x=14\), \(y=-41\times14 + 814=-574 + 814=240\) (not very close), but compared to other options.

Answer:

\(y=-41x + 814\)