A catering company, "Artful Appetizers," provides a price quote for an event based on the number of guests. The company…

The correct answer is 96.

First, determine the linear model for Artful Appetizers's pricing. Let C be the cost and g be the number of guests. The model is C = f + rg, where f is the fixed fee and r is the rate per guest. Using the two data points provided, (20, 650) and (40, 1050), we can calculate the rate per guest (the slope):

r = (1050 - 650) / (40 - 20) = 400 / 20 = 20.

Now, substitute this rate back into the equation with one of the points to find the fixed fee f:

650 = f + 20(20) → 650 = f + 400 → f = 250.

So, the cost model for Artful Appetizers is C_A = 250 + 20g.

The cost model for Better Banquets is given as C_B = 500 + 15g.

Next, find the number of guests at which the two companies' costs are equal:

250 + 20g = 500 + 15g

5g = 250

g = 50.

For fewer than 50 guests, Artful Appetizers is cheaper. For more than 50 guests, Better Banquets is cheaper.

To find the maximum number of guests, we should test the budget constraint with the company that is cheaper for a large number of guests, which is Better Banquets (due to its lower per-guest rate). Let's find the maximum guests possible with Better Banquets within the $1,950 budget:

500 + 15g ≤ 1950

15g ≤ 1450

g ≤ 1450 / 15

g ≤ 96.66...

Since the number of guests must be a whole number, the maximum is 96. At 96 guests, Better Banquets is the cheaper option (since 96 > 50), so this is a valid solution. The cost would be 500 + 15(96) = 500 + 1440 = $1,940, which is within the budget.

For choice A, 85 is incorrect. This is the maximum number of guests if one only used the model for Artful Appetizers: 250 + 20g ≤ 1950 → 20g ≤ 1700 → g ≤ 85. This fails to account for the fact that Better Banquets is the cheaper option for this many guests. For B, 50 is incorrect. This is the number of guests for which the two companies' costs are equal, not the maximum number of guests possible within the budget. For D, 113 is incorrect. This result is obtained by incorrectly combining the lower fixed fee from Artful Appetizers with the lower per-guest rate from Better Banquets: 250 + 15g ≤ 1950 → 15g ≤ 1700 → g ≤ 113.33...