A landscaping company charges a constant amount per hour for lawn-mowing. The total cost for 3 hours of work is $84, and the total cost for 5 hours of work is $140.
Let be the total cost, in dollars, for hours of work. Which equation represents the relationship between and ?
For "constant amount per" word problems, immediately think of a linear equation with form . Find the rate by dividing the total amount by the number of units (here, dollars ÷ hours) using any one example, then quickly check that the same rate works for the other example. Finally, match this rate to the choice written as "total = rate × time" and reject any options where cost decreases with more time or where substituting the given values does not produce the correct totals.
Hints
Identify what is constant
Focus on the phrase "charges a constant amount per hour." How would you write cost as a function of hours if the company charges the same rate every hour?
Use one scenario to find the rate
Use the information that 3 hours costs $84. What operation connects 84 dollars and 3 hours to a "dollars per hour" rate?
Confirm with the other scenario
After you find a candidate hourly rate from the 3-hour case, test it with the 5-hour, $140 case. Does multiplying that rate by 5 give 140?
Match to an equation
Once you know the hourly rate, look for the option where is that rate multiplied by , and where the cost increases (not decreases) as hours increase.
Desmos Guide
Compute and compare the hourly rates
In one line, type 84/3 and in another line, type 140/5. Check that both expressions evaluate to the same number; that common value is the hourly rate in dollars per hour.
Build the cost equation from the rate
Once you see the hourly rate from step 1, write an equation of the form C = (that rate)*h in Desmos. This represents the cost as a function of hours.
Check the given points on the graph
(Optional) Add the points (3,84) and (5,140) in Desmos. Verify that they lie on the line given by your equation; the option whose line goes through both points is the correct one.
Step-by-step Explanation
Translate the situation into a linear model
The company charges a constant amount per hour, so the cost is proportional to the number of hours.
That means the relationship has the form
or, more simply,
- , where is the cost per hour.
Use one data point to find the hourly rate
We know that 3 hours of work costs $84.
Substitute and into :
Solve for :
So the company charges 28 dollars per hour.
Check with the other data point
Now check that this hourly rate also works for 5 hours and $140.
Using and :
This matches the given information, so is consistent with both points.
Write the final equation
Since the hourly rate is 28 dollars per hour and the cost is rate times hours, the equation relating total cost to hours is
So the correct choice is .