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. (4 marks)
b. What is the optimization equation?
Let:
x = number of pastels made
y = number of watercolors made
The profit made from pastels is $40 per pastel, so the profit made from x pastels is 40x.
The profit made from watercolors is $100 per watercolor, so the profit made from y watercolors is 100y.
The material cost for each pastel is $5, so the cost for x pastels is 5x.
The material cost for each watercolor is $15, so the cost for y watercolors is 15y.
Based on the given information:
1. Owen wants to determine a reasonable profit for the month's work, so let's say he wants to earn a profit of P dollars.
2. Owen has $180 to spend on materials.
3. Owen can make at most 16 pictures.
The inequalities to represent this situation are as follows:
1. The profit made should be at least P dollars: 40x + 100y ≥ P
2. The cost of materials should not exceed $180: 5x + 15y ≤ 180
3. Owen can make at most 16 pictures: x + y ≤ 16
Additionally, there is a non-negative constraint:
x ≥ 0 (the number of pastels cannot be negative)
y ≥ 0 (the number of watercolors cannot be negative)
The system of inequalities is:
40x + 100y ≥ P
5x + 15y ≤ 180
x + y ≤ 16
x ≥ 0
y ≥ 0
b. The optimization equation is to maximize the profit made, which can be represented as the objective function:
Maximize Z = 40x + 100y
a. To represent this situation with inequalities, we can define the variables:
Let x represent the number of pastel pictures that Owen makes.
Let y represent the number of watercolor pictures that Owen makes.
Given the following constraints:
1. Each pastel requires $5 in materials, and Owen has $180 to spend on materials. So, the cost of pastel materials should not exceed $180: 5x ≤ 180.
2. Each watercolor requires $15 in materials, and Owen has $180 to spend on materials. So, the cost of watercolor materials should not exceed $180: 15y ≤ 180.
3. Owen can make at most 16 pictures: x + y ≤ 16.
The non-negative constraints are:
x ≥ 0 (the number of pastel pictures cannot be negative)
y ≥ 0 (the number of watercolor pictures cannot be negative)
Thus, the system of inequalities that represents this situation is:
5x ≤ 180
15y ≤ 180
x + y ≤ 16
x ≥ 0
y ≥ 0
b. The optimization equation in this case is to maximize Owen's profit.
The profit from pastel pictures is $40 per picture and the profit from watercolor pictures is $100 per picture. So, the total profit, P, can be calculated as:
P = 40x + 100y