Question

1. the table shows the cost of visiting a working ranch for one day and night for different numbers of people.

number of people	4	6	8	10	12
cost (dollars)	250	350	450	550	650

a) write the regression equation:
b) what is the slope and what does it represent?

1. the table shows the cost of a catered lunch buffet for different numbers of people.

number of people	cost (dollars)
12	192
18	288
24	384
30	480
36	576
42	672

a) find a linear equation that models the data.
b) what does the domain represent?
c) what does the range represent?
d) interpret the rate of change(slope).
e) what is the cost for a lunch buffet for 120 people?

Explanation:

Step1: Find slope for first - table

Use slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(4,250)$ and $(x_2,y_2)=(6,350)$. Then $m=\frac{350 - 250}{6 - 4}=\frac{100}{2}=50$.
Using point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(4,250)$ and $m = 50$, we get $y-250 = 50(x - 4)$, which simplifies to $y=50x + 50$.

Step2: Interpret slope for first - table

The slope $m = 50$ represents the increase in cost (in dollars) per additional person visiting the working ranch.

Step3: Find slope for second - table

Let $(x_1,y_1)=(12,192)$ and $(x_2,y_2)=(18,288)$. Then $m=\frac{288 - 192}{18 - 12}=\frac{96}{6}=16$.
Using point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(12,192)$ and $m = 16$, we get $y-192=16(x - 12)$, which simplifies to $y = 16x$.

Step4: Interpret domain for second - table

The domain represents the number of people for the catered lunch buffet.

Step5: Interpret range for second - table

The range represents the cost (in dollars) of the catered lunch buffet.

Step6: Interpret slope for second - table

The slope $m = 16$ represents the increase in cost (in dollars) per additional person for the catered lunch buffet.

Step7: Find cost for 120 people in second - table

Substitute $x = 120$ into $y = 16x$. So $y=16\times120 = 1920$.

Answer:

1.
a. $y = 50x+50$
b. The slope is 50. It represents the increase in cost (in dollars) per additional person visiting the working ranch.
2.
a. $y = 16x$
b. The domain represents the number of people for the catered lunch buffet.
c. The range represents the cost (in dollars) of the catered lunch buffet.
d. The slope of 16 represents the increase in cost (in dollars) per additional person for the catered lunch buffet.
e. The cost for a lunch buffet for 120 people is 1920 dollars.