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.
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.
f. Owen’s mother has convinced him that he should try to earn as much as possible. So, Owen needs to figure out the most profit he can earn within his constraints. He also wants to be able to prove to his mother that it is really the maximum amount. Find the maximum possible profit that Owen can earn and the combination of pictures he should make to earn that profit.
To find three different combinations of watercolors and pastels that will earn Owen a profit of exactly $500, we need to set up equations based on the given information. Let's assume Owen makes x pastels and y watercolors.
The profit from making x pastels is ($40 * x) and the cost of materials for x pastels is ($5 * x). So, the total profit from pastels is given by the equation: 40x - 5x = 500
Simplifying the equation, we get: 35x = 500
Dividing both sides by 35, we find that x = 500/35 = 14.29
Since Owen can only make whole numbers of pictures, let's try x = 14 and y = 2:
Profit from pastels = $40 * 14 = $560
Profit from watercolors = $100 * 2 = $200
Total profit = $560 + $200 = $760
Since the total profit exceeds $500, this combination does not work.
Now let's try x = 12 and y = 3:
Profit from pastels = $40 * 12 = $480
Profit from watercolors = $100 * 3 = $300
Total profit = $480 + $300 = $780
Again, the total profit exceeds $500, so this combination does not work either.
Next, let's try x = 10 and y = 4:
Profit from pastels = $40 * 10 = $400
Profit from watercolors = $100 * 4 = $400
Total profit = $400 + $400 = $800
This time, the total profit is greater than $500, but not by much. This combination earns Owen exactly $500 in profit.
To find the maximum possible profit that Owen can earn, we can analyze different scenarios:
1. If Owen makes only pastels: Since Owen can make at most 16 pictures, the maximum number of pastels he can make is 16. Therefore, the maximum profit from pastels is $40 * 16 = $640.
2. If Owen makes only watercolors: The maximum number of watercolors Owen can make is 16. Therefore, the maximum profit from watercolors is $100 * 16 = $1600.
3. If Owen makes some combination of pastels and watercolors: From the previous calculations, we found that the combination of x = 10 pastels and y = 4 watercolors yields the maximum profit of $500. So the maximum profit from this combination is $500.
Comparing the profits from the three scenarios, we see that the maximum profit Owen can earn is $1600 from making only watercolors. Thus, Owen should make 16 watercolors to earn the maximum profit.
To find three different combinations of watercolors and pastels that will earn Owen a profit of exactly $500, we can use a trial and error method. Let's start by assuming Owen makes x pastels and y watercolors.
1. Combination 1:
Let's assume Owen makes 5 pastels (x = 5) and 1 watercolor (y = 1).
Profit from pastels = 5 x $40 = $200
Profit from watercolor = 1 x $100 = $100
Total profit = $200 + $100 = $300
This combination does not earn Owen a profit of exactly $500.
2. Combination 2:
Next, let's assume Owen makes 8 pastels (x = 8) and 2 watercolors (y = 2).
Profit from pastels = 8 x $40 = $320
Profit from watercolors = 2 x $100 = $200
Total profit = $320 + $200 = $520
This combination exceeds the profit of $500.
3. Combination 3:
Finally, let's assume Owen makes 7 pastels (x = 7) and 3 watercolors (y = 3).
Profit from pastels = 7 x $40 = $280
Profit from watercolors = 3 x $100 = $300
Total profit = $280 + $300 = $580
This combination exceeds the profit of $500.
Therefore, there are no combinations of watercolors and pastels that will earn Owen a profit of exactly $500.
Now let's move on to finding the maximum possible profit that Owen can earn within his constraints.
To maximize his profit, Owen should focus on making watercolors since they have a higher profit margin per picture.
Let's assume Owen makes x watercolors and y pastels.
For watercolors:
Profit per watercolor = $100
Material cost per watercolor = $15
For pastels:
Profit per pastel = $40
Material cost per pastel = $5
We know:
Material cost constraint: $15x + $5y ≤ $180
Total number of pictures constraint: x + y ≤ 16
We want to maximize: Total profit = $100x + $40y
To find the maximum profit, we can set up a linear programming problem and solve it using optimization techniques. However, based on the given information, we can already deduce the answer.
Let's consider the extreme cases:
If Owen makes all watercolors (x = 16) and no pastels (y = 0):
Profit = $100 x 16 = $1600
Material cost = $15 x 16 = $240 (within constraint)
If Owen makes all pastels (x = 0) and no watercolors (y = 16):
Profit = $40 x 16 = $640
Material cost = $5 x 16 = $80 (within constraint)
Therefore, the maximum possible profit that Owen can earn is $1600, and he should make 16 watercolors and no pastels to achieve this profit.