A fixed fee plus a constant price per item forms a linear model: total cost equals the fee plus the per-item price times the number ordered. Use the two orders as input-output facts, then fit one rule with a matched-list regression in Desmos. Don't mistake the total for an order for the fee; the fee is the cost at zero items.
Hints
- Hint 1
In a linear model, a constant price per booklet gets multiplied by the number of booklets, while a one-time fee gets added once. How can you write the total using an unknown price and an unknown fee?
- Hint 2
Each order gives an input-output pair: the booklet count is the input, and the amount paid is the output. Substitute each pair into your cost rule to get two equations.
- Hint 3
A regression in Desmos can fit both equations at once. Put the two known totals in one list and their matching cost expressions in another, keeping the pairs in the same order.
Step-by-step
Fit the fee and per-booklet price
Step 1Write a rule for the two charges
The cost per booklet is constant, so it contributes an amount for every booklet. The fixed design fee is charged once per order. Let be the dollars per booklet and be the design fee. Then the linear model is .
- Step 2
Turn the orders into equations
Put each booklet count and its matching total into the rule:
Both equations use the same fee, . An order's total is not the fee when that order contains booklets.
- Step 3
Fit both equations in Desmos
Type . The regression symbol tells Desmos to find one and one that match both totals. Under PARAMETERS, Desmos shows and . That's dollars per booklet and a -dollar design fee.
- Step 4
Write the total-cost function
Put the per-booklet price and fee into :
So the company charges for each booklet plus a one-time design fee. Choice D.