Owen would like to make a small income as an artist. Owen asked his friend Emily for advice about what combination of pictures to make. She suggested that he determine a reasonable profit for that month’s work
and then paint what he needs in order to earn that amount of profit.
• Each pastel requires $5 in materials and earns a profit of $40 for Owen.
• Each watercolor requires $15 in materials and earns a profit of $100 for Owen.
• Owen has $180 to spend on materials.
• Owen can make at most 16 pictures.
a. State the system of inequalities that represents this situation. Remember to define your variables and include any non-negative constraints that are required.
Let x be the number of pastel pictures Owen makes, and let y be the number of watercolor pictures Owen makes.
The inequalities representing this situation are:
5x + 15y ≤ 180 (constraint on the amount of money Owen can spend on materials)
x + y ≤ 16 (constraint on the maximum number of pictures Owen can make)
In addition, both x and y must be non-negative:
x ≥ 0
y ≥ 0
Therefore, the system of inequalities is:
5x + 15y ≤ 180
x + y ≤ 16
x ≥ 0
y ≥ 0
Let's define the variables:
Let x be the number of pastel pictures Owen makes.
Let y be the number of watercolor pictures Owen makes.
We can set up the following system of inequalities:
1. Cost constraint: 5x + 15y ≤ 180 (The total cost of materials cannot exceed $180).
2. Profit constraint: 40x + 100y ≥ P (The total profit Owen earns should be at least P).
3. Non-negative constraint: x ≥ 0 (Owen cannot make a negative number of pastel pictures).
4. Non-negative constraint: y ≥ 0 (Owen cannot make a negative number of watercolor pictures).
5. Total number of pictures constraint: x + y ≤ 16 (Owen can make at most 16 pictures).
P represents the target profit Owen wants to earn for that month's work.