Let's analyze the situation in detail based on the given data and numerical results.
### Step 1: Determine the Monopolist's Profit-Maximizing Price and Quantity
A monopoly aims to maximize its total revenue, which is the product of the price per gallon and the quantity demanded.
Given data:
- Price per gallon: [6.00, 5.50, 5.00, 4.50, 4.00, 3.50, 3.00, 2.50, 2.00, 1.50, 1.00, 0.50, 0]
- Quantity demanded: [0, 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 540, 0]
- Total Revenue: [0, 247.50, 450.00, 607.50, 720.00, 787.50, 810.00, 787.50, 450.00, 247.50, 0, None, None]
To find the profit-maximizing price and quantity, we need to identify where the total revenue is maximized.
### Step 2: Maximize Total Revenue
The maximum total revenue from the given data is [tex]$810.00. This occurs at a price of $[/tex]3.00 per gallon and a quantity of 270 gallons.
So, the profit-maximizing price (s) is [tex]$3.00 per gallon, and the total output is 270 gallons.
### Step 3: Split Production Equally
Isaiah and Dalia agree to split the production equally. Therefore, each will produce half of the monopoly quantity.
Isaiah's production: \( \frac{270}{2} = 135 \) gallons
Dalia's production: \( \frac{270}{2} = 135 \) gallons
Isaiah's profit (since price per gallon times quantity = revenue):
\[ 135 \text{ gallons} \times 3 \text{ dollars per gallon} = \$[/tex]405 \]
Dalia's profit:
[tex]\[ 135 \text{ gallons} \times 3 \text{ dollars per gallon} = \$405 \][/tex]
### Step 4: Isaiah Increases Production by 4 Gallons
Now, Isaiah decides to increase his production by 4 gallons. Isaiah's new production level will be:
[tex]\[ 135 + 4 = 139 \text{ gallons} \][/tex]
Total new quantity after Isaiah’s increase:
[tex]\[ 139 \text{ gallons} + 135 \text{ gallons} = 274 \text{ gallons} \][/tex]
### Step 5: Determine New Market Price
The new market quantity is 274 gallons. To find the new market price, we need to interpolate between the quantities and their corresponding prices. For the sake of simplicity, the new price approximates to a little less than [tex]$3.00 per gallon due to increased quantity:
New market price: approximately $[/tex]2.956 per gallon
### Step 6: Calculate New Profits
Isaiah’s new profit using the new market price:
[tex]\[ 139 \text{ gallons} \times 2.956 \text{ dollars per gallon} = \$410.82 \][/tex]
Dalia’s new profit using the new market price:
[tex]\[ 135 \text{ gallons} \times 2.956 \text{ dollars per gallon} = \$399 \][/tex]
So, summarizing:
1. Profit-maximizing price (s): [tex]$3.00 per gallon
2. Total output: 270 gallons
3. Isaiah's initial profit: \$[/tex]405
4. Dalia's initial profit: \[tex]$405
5. New total quantity after Isaiah’s increase: 274 gallons
6. New price per gallon: approximately $[/tex]2.956
7. Isaiah's new profit: \[tex]$410.82
8. Dalia's new profit: \$[/tex]399
