A fixed fee plus a charge per guest is a linear model: total cost equals the per-guest rate times the guest count, plus the fee. Two guest-count and cost pairs let you fit both numbers with a Desmos table regression. Don’t trust a rule that matches only one banquet; it must match both.
Hints
- Hint 1
In a linear model, the per-guest charge multiplies the number of guests, while the fixed fee is added once. What form should have if the rate and fee are still unknown?
- Hint 2
Each guest count and its total cost form an input-output pair. Put the two pairs in a Desmos table, keeping each cost on the same row as its guest count.
- Hint 3
A regression fits a rule to the table. Type to find a rate and a fee that match both rows.
Step-by-step
Fit the per-guest rate and setup fee
Step 1Write the cost model
A linear model adds the same amount for each new guest. Let be the cost per guest and the fixed setup fee, so . The setup fee is added once, not multiplied by .
- Step 2
Translate the two banquets
The guest counts and costs give two conditions: . One cost alone can’t determine both and , so you need both.
- Step 3
Enter the cost pairs
Enter a Desmos table with for guests and for cost. It shows beside and beside . Keep each price on the same row as its guest count.
- Step 4
Fit both parts of the cost
Type . The regression uses both rows to find one rate and one fee. Under PARAMETERS, Desmos shows and : that’s $9 per guest and a $200 setup fee.
- Step 5
Give the total-cost function
Put the per-guest charge beside and add the setup fee: . Multiply the per-guest rate by the guest count; add the fixed fee only once. Choice B.