A company produces mountain-bikes and city-bikes. The demand function of the mountain-bikes is given by the equation: Q1 = 367.5 - 0.5₽1
Pi is the price per mountain-bike in euros. Qi is the number of mountain-bikes demanded per month.
The demand function of the city-bikes is given by the equation: Q2 = 600 - P2
Pa is the price per city-bike in euros. Q2 is the number of city-bikes demanded per month.
Total costs in euros for producing Q1 mountain-bikes and Q2 city-bikes:
TC (Q1, Q2) = 3Q1hoch 2 + Q1Q2+Q2 hoch 2+540.
Compute the number of mountain-bikes and city-bikes for which the monthly profit is maximum and the maximum profit.
Question: What is the maximum profit? Round to an integer amount of euros.

To determine the number of mountain-bikes (\( Q_1 \)) and city-bikes (\( Q_2 \)) that maximize profit and calculate the maximum profit, we need to find the profit function and then optimize it.

### Step 1: Define the Revenue and Cost Functions

1. **Demand Functions**:
   - Mountain-bikes: \( Q_1 = 367.5 - 0.5P_1 \)
   - City-bikes: \( Q_2 = 600 - P_2 \)

   Rearranging to express prices in terms of quantities:
   - \( P_1 = 735 - 2Q_1 \)
   - \( P_2 = 600 - Q_2 \)

2. **Revenue Function**:
   - Revenue from mountain-bikes: \( R_1 = P_1 \cdot Q_1 = (735 - 2Q_1) \cdot Q_1 = 735Q_1 - 2Q_1^2 \)
   - Revenue from city-bikes: \( R_2 = P_2 \cdot Q_2 = (600 - Q_2) \cdot Q_2 = 600Q_2 - Q_2^2 \)
   - Total Revenue: \( R = R_1 + R_2 = 735Q_1 - 2Q_1^2 + 600Q_2 - Q_2^2 \)

3. **Total Cost Function**:
   - Given: \( TC(Q_1, Q_2) = 3Q_1^2 + Q_1Q_2 + Q_2^2 + 540 \)

4. **Profit Function**:
   - Profit: \( \pi(Q_1, Q_2) = R - TC = (735Q_1 - 2Q_1^2 + 600Q_2 - Q_2^2) - (3Q_1^2 + Q_1Q_2 + Q_2^2 + 540) \)
   - Simplify the profit function:
     \[
     \pi(Q_1, Q_2) = 735Q_1 - 5Q_1^2 + 600Q_2 - 2Q_2^2 - Q_1Q_2 - 540
     \]

### Step 2: Find the Critical Points

To maximize profit, we take the partial derivatives of the profit function with respect to \( Q_1 \) and \( Q_2 \), and set them to zero.

1. **Partial Derivative with respect to \( Q_1 \)**:
   \[
   \frac{\partial \pi}{\partial Q_1} = 735 - 10Q_1 - Q_2 = 0
   \]

2. **Partial Derivative with respect to \( Q_2 \)**:
   \[
   \frac{\partial \pi}{\partial Q_2} = 600 - 4Q_2 - Q_1 = 0
   \]

Solving these equations simultaneously:

From the first equation:
\[
Q_2 = 735 - 10Q_1
\]

Substitute \( Q_2 \) into the second equation:
\[
600 - 4(735 - 10Q_1) - Q_1 = 0
\]

Simplify:
\[
600 - 2940 + 40Q_1 - Q_1 = 0
\]
\[
39Q_1 = 2340
\]
\[
Q_1 = 60
\]

Now, substitute \( Q_1 = 60 \) into the equation for \( Q_2 \):
\[
Q_2 = 735 - 10 \times 60 = 135
\]

### Step 3: Calculate Maximum Profit

Substitute \( Q_1 = 60 \) and \( Q_2 = 135 \) into the profit function:
\[
\pi(60, 135) = 735(60) - 5(60)^2 + 600(135) - 2(135)^2 - 60(135) - 540
\]

Simplify step-by-step:
\[
\pi(60, 135) = 44100 - 18000 + 81000 - 36450 - 8100 - 540
\]
\[
\pi(60, 135) = 126000 - 45090
\]
\[
\pi(60, 135) = 80810
\]

### Final Answer

The maximum profit is **€80,810**. 

Would you like more details, or do you have any questions?

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Here are some related questions to expand your understanding:

1. How would the profit change if the cost function were different?
2. What if the demand for city-bikes increased? How would it affect the optimal production?
3. How sensitive is the profit to changes in the price of mountain-bikes?
4. How would fixed costs influence the overall profit?
5. What is the economic interpretation of the second derivatives of the profit function?

**Tip**: Analyzing how changes in costs or demand functions affect profits can give insights into business strategy adjustments!

A company produces mountain-bikes and city-bikes. The demand function of the mountain-bikes is given by the equation: Q1 = 367.5 - 0.5₽1. Pi is the price per mountain-bike in euros. Qi is the number of mountain-bikes demanded per month. The demand function of the city-bikes is given by the equation: Q2 = 600 - P2. Pa is the price per city-bike in euros. Q2 is the number of city-bikes demanded per month. Total costs in euros for producing Q1 mountain-bikes and Q2 city-bikes: TC (Q1, Q2) = 3Q1^2 + Q1Q2 + Q2^2 + 540. Compute the number of mountain-bikes and city-bikes for which the monthly profit is maximum and the maximum profit. What is the maximum profit? Round to an integer amount of euros.

Learn how to maximize monthly profit for a company producing mountain-bikes and city-bikes using demand and cost functions. This detailed solution covers key mathematical concepts like optimization, quadratic functions, and partial derivatives. Suitable for undergraduate students studying business math or economics.

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