Sure! Let's break down the problem step-by-step to find the profit-maximizing quantity and the corresponding profit for a monopoly.
### Step-by-Step Solution:
1. List of Quantities, Prices, and Total Costs:
- Quantity: [tex]\(0, 1, 2, 3, 4, 5\)[/tex]
- Price: [tex]\(\$32, \$28, \$24, \$20, \$16, \$12\)[/tex]
- Total Cost: [tex]\(\$6, \$20, \$34, \$48, \$62, \$76\)[/tex]
2. Compute Total Revenue:
Total Revenue for each quantity is given by:
[tex]\[
\text{Total Revenue} = \text{Quantity} \times \text{Price}
\][/tex]
So, we calculate it for each quantity:
- [tex]\(0 \times 32 = 0\)[/tex]
- [tex]\(1 \times 28 = 28\)[/tex]
- [tex]\(2 \times 24 = 48\)[/tex]
- [tex]\(3 \times 20 = 60\)[/tex]
- [tex]\(4 \times 16 = 64\)[/tex]
- [tex]\(5 \times 12 = 60\)[/tex]
Thus, Total Revenues are: [tex]\([0, 28, 48, 60, 64, 60]\)[/tex]
3. Compute Profit:
Profit for each quantity is given by:
[tex]\[
\text{Profit} = \text{Total Revenue} - \text{Total Cost}
\][/tex]
We calculate it for each quantity:
- [tex]\(0 - 6 = -6\)[/tex]
- [tex]\(28 - 20 = 8\)[/tex]
- [tex]\(48 - 34 = 14\)[/tex]
- [tex]\(60 - 48 = 12\)[/tex]
- [tex]\(64 - 62 = 2\)[/tex]
- [tex]\(60 - 76 = -16\)[/tex]
Thus, Profits are: [tex]\([-6, 8, 14, 12, 2, -16]\)[/tex]
4. Identify the Maximum Profit:
We look at the calculated profits and see which one is the highest:
- Maximum profit is [tex]\(14\)[/tex], which occurs at a quantity of [tex]\(2\)[/tex].
5. Conclusion:
At the profit-maximizing quantity of [tex]\(2\)[/tex], the monopoly will earn a profit of [tex]\(\$ 14\)[/tex].
Therefore, the answer to the problem is:
[tex]\(\boxed{14}\)[/tex]
