Question

the table shows the numbers of ice - cream cones sold for different outside temperatures (in degrees fahrenheit).
temperature, x: 63 62 70 82 80
number of cones, y: 90 105 117 131 147
do the data show a linear relationship?
yes
no
if the data show a linear relationship, write an equation of a line of fit using the ordered pairs (62,105) and (70,117). then use the equation to estimate how many ice - cream cones are sold when the temperature is 68°f. if the data do not show a linear relationship, leave the following blank.
an equation is y = ( )

Explanation:

Step1: Calculate the slope

The slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $m=\frac{y_2 - y_1}{x_2 - x_1}$. Using the points $(62,105)$ and $(70,117)$, we have $x_1 = 62$, $y_1=105$, $x_2 = 70$, $y_2 = 117$. Then $m=\frac{117 - 105}{70 - 62}=\frac{12}{8}=1.5$.

Step2: Find the y - intercept

We use the point - slope form of a line $y - y_1=m(x - x_1)$ and then convert it to the slope - intercept form $y=mx + b$. Using the point $(62,105)$ and $m = 1.5$, we have $y-105=1.5(x - 62)$. Expand the right side: $y-105=1.5x-93$. Add 105 to both sides to get $y=1.5x + 12$.

Step3: Check linearity by observing data trend

As the temperature $x$ increases from 62 to 70 to 82 to 90, the number of cones $y$ increases from 105 to 117 to 131 to 147 in an approximately linear fashion. So the data shows a linear relationship.

Step4: Predict the number of cones at $x = 68$

Substitute $x = 68$ into the equation $y=1.5x + 12$. Then $y=1.5\times68+12=102 + 12=114$.

Answer:

Yes
$y = 1.5x+12$
114