A linear model changes by the same number of dollars for each extra gigabyte. Two customer records give you that rate: calculate it in Desmos, build the bill from one actual record, then solve backward from the given charge. Keep each bill paired with its own usage. The endpoints of the allowed range are not extra customer records.
Hints
- Hint 1
An ordered pair puts the input first and the output second. Here, usage is the input and the bill is the output, so write each customer’s record with gigabytes first. What are the two points?
- Hint 2
The slope is the change in the bill for one extra gigabyte. Divide the difference between the two bills by the difference between their matching usages, subtracting in the same order.
- Hint 3
Start with one customer’s bill, then add the slope times the change in usage from that customer. The question gives you a bill, not a usage. What input makes your model produce that bill?
Step-by-step
Find the rate, then solve for usage
Step 1Pair each usage with its bill
In an ordered pair, the input comes first. Usage is the input and bill is the output, so the customer records give and . Keep 6 with 45 and 14 with 85; the 5-GB and 15-GB limits aren’t customer records.
- Step 2
Find the dollars per gigabyte
The slope is the bill change divided by the matching usage change. Type into Desmos; it prints . So the bill rises $5 per GB. Dividing by instead would assume a $0 bill at zero usage, which the problem doesn’t say.
- Step 3
Build the bill from a known customer
Use point-slope form: start with the $45 bill at 6 GB, then add the rate times the change from 6 GB. That gives . The keeps $45 tied to 6 GB, not to the 5-GB lower limit.
- Step 4
Solve backward from the bill
The customer paid $67, so set the model equal to that output. Type in Desmos. Here is the unknown usage, and tells Desmos to find its value. Under PARAMETERS, Desmos shows . That lies between 5 and 15 GB, so the customer used 10.4 gigabytes. Choice B.