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 represent the number of pastels and y represent the number of watercolors that Owen makes.
The system of inequalities representing this situation can be written as:
5x + 15y ≤ 180 (constraint on materials cost)
x + y ≤ 16 (constraint on the number of pictures)
x ≥ 0, y ≥ 0 (non-negative constraints)
b. The optimization equation is to maximize the profit, which can be represented as:
Profit = 40x + 100y
d. To find three different combinations of watercolors and pastels that would earn Owen a profit of exactly $1,000, let's substitute the profit value into the optimization equation and solve the system of inequalities:
Profit = 40x + 100y = 1000
First combination:
40x + 100y = 1000
Solving with the constraints, we can substitute x = 2 and y = 8 to get:
40(2) + 100(8) = 80 + 800 = $880 profit
Second combination:
40x + 100y = 1000
Solving with the constraints, we can substitute x = 4 and y = 6 to get:
40(4) + 100(6) = 160 + 600 = $760 profit
Third combination:
40x + 100y = 1000
Solving with the constraints, we can substitute x = 6 and y = 4 to get:
40(6) + 100(4) = 240 + 400 = $640 profit
a.
Let's define the variables for this problem:
Let x be the number of pastel pictures Owen makes.
Let y be the number of watercolor pictures Owen makes.
According to the given conditions, we can write the following inequalities:
1. The cost of materials used for pastel pictures:
5x ≤ 180
2. The cost of materials used for watercolor pictures:
15y ≤ 180
3. Owen can make at most 16 pictures:
x + y ≤ 16
4. We need to determine a reasonable profit of $1,000:
40x + 100y ≥ 1000
Since the variables must be non-negative, we also need to include the following constraint:
x, y ≥ 0
b.
The optimization equation is to maximize the profit, which can be represented as the objective function:
P = 40x + 100y
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:
40x + 100y = 1000
5x + 15y ≤ 180
x + y ≤ 16
x, y ≥ 0
We can use a method like graphing or substitution to find the solutions for x and y.