Question

1. a caterer charges a setup fee and a flat amount per person served. if 45 people are served, the total bill will be $654. if 68 people are served, the total bill will be $953.

a. write an equation in slope - intercept form that gives the cost of serving x people.

b. what are the slope and y - intercept, and what do they mean?

c. how many people could be served for $1100? explain how you found your answer.

Explanation:

Step1: Define variables and find slope

Let \( x \) be the number of people, \( y \) be the total cost. Two points: \((45, 654)\) and \((68, 953)\). Slope \( m=\frac{953 - 654}{68 - 45}=\frac{299}{23}=13\).

Step2: Find y - intercept

Use point - slope form \( y - y_1=m(x - x_1)\) with \((45, 654)\): \( y-654 = 13(x - 45)\). Expand: \( y-654=13x-585\). Then \( y = 13x+69\).

Step3: Interpret slope and y - intercept

Slope \( m = 13\): Cost per person is \(\$13\). Y - intercept \( b = 69\): Setup fee is \(\$69\).

Step4: Solve for x when y = 1100

Set \( 1100=13x + 69\). Subtract 69: \( 1031 = 13x\). Then \( x=\frac{1031}{13}\approx79.31\), so 79 people (since we can't serve a fraction of a person).

Part (a)

Slope \( m = 13 \): Represents the flat amount (in dollars) charged per person served. Y - intercept \( b = 69 \): Represents the setup fee (in dollars) charged regardless of the number of people.

Use the equation \( y = 13x+69 \). Set \( y = 1100 \): \( 1100=13x + 69 \). Subtract 69: \( 13x=1100 - 69 = 1031 \). Then \( x=\frac{1031}{13}\approx79.31\). Since we can't serve a fraction of a person, we take the whole number part, so 79 people.

Answer:

\( y = 13x+69 \)

Part (b)