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.
b. What is the optimization equation?
d. Suppose Owen decided $1,000 would be a reasonable profit. Find three different combinations of
watercolors and pastels that would earn Owen a profit of exactly $1,000. (3 marks)
e. Now suppose Owen wanted to earn only $500 in profit. Find three different combinations of
watercolors and pastels that will earn Owen a profit of exactly $500. Using a different-coloured
pencil, add those points to your graph.
a. Let:
x = the number of pastels Owen makes
y = the number of watercolors Owen makes
The system of inequalities is:
5x + 15y ≤ 180 (constraint for available materials)
x + y ≤ 16 (constraint for number of pictures)
Additionally, the constraints must be non-negative:
x ≥ 0
y ≥ 0
b. The optimization equation is:
Profit = 40x + 100y
d. To find three different combinations of watercolors and pastels that would earn Owen a profit of exactly $1,000, we must solve the system of inequalities along with the optimization equation:
5x + 15y ≤ 180
x + y ≤ 16
x ≥ 0
y ≥ 0
Profit = 40x + 100y = 1000
One possible solution is:
x = 4
y = 8
Another possible solution is:
x = 2
y = 10
And one more possible solution is:
x = 6
y = 7
e. To find three different combinations of watercolors and pastels that will earn Owen a profit of exactly $500, we need to solve the same system of inequalities and optimization equation, but change the profit value to 500:
5x + 15y ≤ 180
x + y ≤ 16
x ≥ 0
y ≥ 0
Profit = 40x + 100y = 500
One possible solution is:
x = 3
y = 6
Another possible solution is:
x = 0
y = 5
And one more possible solution is:
x = 6
y = 2
These points can be added to the graph as requested.
a. Let's define the variables:
Let x represent the number of pastel paintings Owen makes.
Let y represent the number of watercolor paintings Owen makes.
The constraints in this situation are:
1. Each pastel painting requires $5 in materials and earns a profit of $40 for Owen, so the cost of pastel paintings is 5x and the profit from pastel paintings is 40x.
2. Each watercolor painting requires $15 in materials and earns a profit of $100 for Owen, so the cost of watercolor paintings is 15y and the profit from watercolor paintings is 100y.
3. Owen has $180 to spend on materials, so the total cost of materials must not exceed $180: 5x + 15y ≤ 180.
4. Owen can make at most 16 pictures, so the total number of pictures must not exceed 16: x + y ≤ 16.
5. The number of pastel paintings and watercolor paintings must be non-negative: x ≥ 0 and y ≥ 0.
Therefore, the system of inequalities representing this situation is:
5x + 15y ≤ 180
x + y ≤ 16
x ≥ 0
y ≥ 0
b. The optimization equation is to maximize the profit, which can be expressed as:
Profit = 40x + 100y
d. To find three different combinations of watercolors and pastels that would earn Owen a profit of exactly $1,000, we can solve the system of equations created by the constraints and the optimization equation:
5x + 15y ≤ 180 (cost constraint)
x + y ≤ 16 (total number of pictures constraint)
Profit = 40x + 100y = $1,000
One possible combination is:
x = 3 (number of pastel paintings)
y = 8 (number of watercolor paintings)
Another possible combination is:
x = 6 (number of pastel paintings)
y = 6 (number of watercolor paintings)
And a third possible combination is:
x = 9 (number of pastel paintings)
y = 4 (number of watercolor paintings)
e. To find three different combinations of watercolors and pastels that will earn Owen a profit of exactly $500, we can solve the system of equations created by the constraints and the optimization equation:
5x + 15y ≤ 180 (cost constraint)
x + y ≤ 16 (total number of pictures constraint)
Profit = 40x + 100y = $500
One possible combination is:
x = 1 (number of pastel paintings)
y = 5 (number of watercolor paintings)
Another possible combination is:
x = 3 (number of pastel paintings)
y = 4 (number of watercolor paintings)
And a third possible combination is:
x = 6 (number of pastel paintings)
y = 2 (number of watercolor paintings)