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:
Profit = 40x + 100y = 500
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:
Profit = 40x + 100y = 500
Now, we can find three different combinations by choosing different values for x and y that satisfy this equation. Let's find one possible solution:
Let x = 5 and y = 1:
Profit = 40(5) + 100(1) = 200 + 100 = $300
So, the combination of 5 watercolors and 1 pastel will earn Owen a profit of $300.
Let's find another possible solution:
Let x = 8 and y = 3:
Profit = 40(8) + 100(3) = 320 + 300 = $620
So, the combination of 8 watercolors and 3 pastels will earn Owen a profit of $620.
Let's find the third possible solution:
Let x = 10 and y = 0:
Profit = 40(10) + 100(0) = 400 + 0 = $400
So, the combination of 10 watercolors and 0 pastels will earn Owen a profit of $400.
Therefore, three different combinations of watercolors and pastels that will earn Owen a profit of exactly $500 are:
1) 5 watercolors and 1 pastel (profit of $300)
2) 8 watercolors and 3 pastels (profit of $620)
3) 10 watercolors and 0 pastels (profit of $400)
To find three different combinations of watercolors and pastels that will earn Owen a profit of exactly $500, we can solve the equation:
40x + 100y = 500,
where x represents the number of watercolors and y represents the number of pastels.
We also need to consider the given constraints:
x >= 0 (non-negativity constraint),
y >= 0 (non-negativity constraint),
4x + 2y <= 15 (time constraint),
2x + 5y <= 15 (material constraint).
To solve this system of inequalities and optimization equation, we can use a method called linear programming.
Let's go through the steps to find three different combinations:
Step 1: Graph the feasible region:
Plot the lines 4x + 2y = 15 and 2x + 5y = 15.
Mark the feasible region, which is the region that satisfies all the constraints.
Step 2: Identify the corner points:
Find the coordinates of the corner points of the feasible region by solving the system of equations formed by the lines that bound the feasible region.
Step 3: Calculate the profit at each corner point:
Plug in the x and y values of each corner point into the profit equation, 40x + 100y = 500, to find the profit at each combination.
Step 4: Select three combinations with a profit of exactly $500:
Choose three combinations with a profit of exactly $500 from the values obtained in Step 3.
Following these steps will help us find three different combinations of watercolors and pastels that will earn Owen a profit of exactly $500.