Roberto is having his house painted. The job takes three days, and he pays the painter the same hourly rate every day. The cost of the job is in the chart below.

Question

Day
$$1$$ Day
$$2$$ Day
$$3$$
Hours worked
$$5$$
$$4$$
$$6$$
Amount paid
$$\mathrm{\$}300$$
$$\mathrm{\$}240$$
$$\mathrm{\$}360$$
What is the painter's unit rate of change of dollars with respect to time; that is, how much is the painter paid for one hour worked?
The unit rate of change is
dollars per hour.
Graph the proportional relationship described above, with the
$$x$$-coordinate representing hours worked, and the
$$y$$-coordinate representing amount paid in dollars.
$$x$$
$$y$$
Graph the relationship.

Answer 1

To find the painter's unit rate of change, we first need to determine how much the painter is paid per hour.

We have the total amount paid and the hours worked for each day:

• Day 1: 5 hours worked, $300 paid
• Day 2: 4 hours worked, $240 paid
• Day 3: 6 hours worked, $360 paid

Step 1: Calculate the Hourly Rate

We can verify that the hourly rate is constant by dividing the total amount paid for each day by the hours worked on that day.

1. Day 1: $$\text{Hourly rate}=\frac{\$300}{5\text{ hours}}=\$60\text{ per hour}$$

2. Day 2: $$\text{Hourly rate}=\frac{\$240}{4\text{ hours}}=\$60\text{ per hour}$$

3. Day 3: $$\text{Hourly rate}=\frac{\$360}{6\text{ hours}}=\$60\text{ per hour}$$

Since the hourly rate is the same for all three days, we can conclude that the unit rate of change is:

$$\text{Unit rate}=\$60\text{ per hour}$$

Step 2: Prepare the Data for Graphing

We can now construct the (x, y) pairs for graphing where $x$ represents hours worked and $y$ represents amount paid.

• Day 1: $(5,300)$
• Day 2: $(4,240)$
• Day 3: $(6,360)$

Step 3: Graph the Relationship

1. Plot the points on a coordinate plane:

   • Point 1: $(5,300)$
   • Point 2: $(4,240)$
   • Point 3: $(6,360)$

2. Draw a line through the points to represent the linear relationship between hours worked and the amount paid.

Observing the Relationship

The points will fall on a straight line due to the consistent hourly wage, confirming that the relationship is proportional.

Summary

• The painter's unit rate of change is $\$60$ per hour.
• The plotted points on the graph will show a straight line reflecting the linear relationship between hours worked and dollar amount paid.