A linear model fits a cost that changes by the same amount for each extra hour. Turn the given totals into (hours, dollars) pairs, then divide the change in cost by the change in hours to find the hourly rate. Use one total to check for a fixed fee before writing the equation; a constant hourly rate alone doesn't rule one out.
Hints
- Hint 1
An input-output pair puts the number of hours first and the total cost second. Write a pair for each given total. How much do the hours and the cost change between the two pairs?
- Hint 2
The hourly rate is the change in dollars divided by the change in hours. Keep the two subtractions in the same order, then evaluate the quotient. The cost increase alone is not a per-hour amount.
- Hint 3
The starting cost is the charge at . Once you know the hourly rate, subtract the charge for hours from the given -hour total. Is there a fixed fee left?
Step-by-step
Find the hourly rate and starting cost
Step 1Turn the totals into pairs
Write each input-output pair as : and . Hours go first because is the number of hours and is the cost.
- Step 2
Find the hourly rate
The hourly rate is the change in cost divided by the change in hours. Keep the pairs in the same order:
Type in Desmos. It shows , so the rate is dollars per hour. The $56 increase took hours, not .
- Step 3
Use a known cost to check for a fee
The starting cost is the charge at hours. With an hourly rate of , the cost rule has the form . Apply the given -hour total:
Don't assume from the words “per hour”; check it.
- Step 4
Find the starting cost
Subtract the charge for hours from both sides:
Add in Desmos. It shows , so there is no fixed fee.
- Step 5
Write the cost equation
Put the rate and starting cost into the linear model :
Drop the zero:
The total cost is $28 per hour with no added fee. Choice B.