1. 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.
a. Let x be the number of pastels Owen makes and y be the number of watercolors Owen makes.
The cost of materials for the pastels is 5x.
The cost of materials for the watercolors is 15y.
Owen's total budget for materials is $180.
The profit from each pastel is $40, so the profit from x pastels is 40x.
The profit from each watercolor is $100, so the profit from y watercolors is 100y.
Owen can make at most 16 pictures, so x + y ≤ 16.
To determine the reasonable profit for the month, Owen needs to earn $1,000.
The optimization equation is 40x + 100y = 1000.
b. The optimization equation is 40x + 100y = 1000.
d. To earn a profit of exactly $1,000, we need to solve the optimization equation for three different combinations of x and y.
First Combination:
Let's set x = 5 and y = 6.
40(5) + 100(6) = 200 + 600 = $800.
This combination will earn Owen a profit of $800.
Second Combination:
Let's set x = 10 and y = 2.
40(10) + 100(2) = 400 + 200 = $600.
This combination will earn Owen a profit of $600.
Third Combination:
Let's set x = 0 and y = 10.
40(0) + 100(10) = 0 + 1000 = $1000.
This combination will earn Owen a profit of $1000.
a. Let's define the variables:
Let x be the number of pastels that Owen makes.
Let y be the number of watercolors that Owen makes.
The system of inequalities that represents this situation is:
5x + 15y ≤ 180 (the total cost of materials must be less than or equal to $180)
x + y ≤ 16 (the total number of pictures cannot exceed 16)
x ≥ 0 (the number of pastels must be non-negative)
y ≥ 0 (the number of watercolors must be non-negative)
b. The optimization equation is:
Profit = 40x + 100y (the total profit Owen earns from selling the pictures)
d. To find three different combinations of watercolors and pastels that would earn Owen a profit of exactly $1,000, we need to solve the system of equations:
5x + 15y = 180 (equation for the cost of materials)
40x + 100y = 1000 (equation for the desired profit)
One possible combination is:
x = 2 (2 pastels)
y = 8 (8 watercolors)
Another combination is:
x = 6 (6 pastels)
y = 4 (4 watercolors)
And another combination is:
x = 10 (10 pastels)
y = 0 (0 watercolors)