https://file.aiquickdraw.com/mathbot/file/1728268457217o1svfxtz.jpg

The image presents a question about a company's cost function and demand function. Let's solve it step by step:

### Given:
- **Total Cost (TC) Function**: \( TC = 50 + 3Q \)
- **Demand Function**: \( P = 100 - 2Q \)

Where \( Q \) is the quantity produced and sold, and \( P \) is the price.

### a. **Total Revenue (TR) Function:**
Total revenue is calculated by multiplying the price per unit (P) by the quantity sold (Q).

\[
TR = P \times Q
\]

Substitute the price function \( P = 100 - 2Q \) into the revenue function:

\[
TR = (100 - 2Q) \times Q
\]
\[
TR = 100Q - 2Q^2
\]

So, the total revenue function is:

\[
TR(Q) = 100Q - 2Q^2
\]

### b. **Profit Function:**
Profit (\( \pi \)) is calculated as total revenue minus total cost:

\[
\pi(Q) = TR(Q) - TC(Q)
\]

Substitute the total revenue and total cost functions:

\[
\pi(Q) = (100Q - 2Q^2) - (50 + 3Q)
\]
\[
\pi(Q) = 100Q - 2Q^2 - 50 - 3Q
\]
\[
\pi(Q) = 97Q - 2Q^2 - 50
\]

Thus, the profit function is:

\[
\pi(Q) = 97Q - 2Q^2 - 50
\]

### c. **Breakeven Point:**
The breakeven point occurs when profit is zero, i.e., when total revenue equals total cost:

\[
TR(Q) = TC(Q)
\]

From the total revenue function and total cost function:

\[
100Q - 2Q^2 = 50 + 3Q
\]
Rearrange the equation:

\[
-2Q^2 + 100Q - 3Q - 50 = 0
\]
\[
-2Q^2 + 97Q - 50 = 0
\]

This is a quadratic equation in the form \( ax^2 + bx + c = 0 \). We can solve it using the quadratic formula:

\[
Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \( a = -2 \), \( b = 97 \), and \( c = -50 \):

\[
Q = \frac{-97 \pm \sqrt{97^2 - 4(-2)(-50)}}{2(-2)}
\]
\[
Q = \frac{-97 \pm \sqrt{9409 - 400}}{-4}
\]
\[
Q = \frac{-97 \pm \sqrt{9009}}{-4}
\]
\[
Q = \frac{-97 \pm 94.95}{-4}
\]

Now calculate both roots:

1. \( Q_1 = \frac{-97 + 94.95}{-4} = \frac{-2.05}{-4} = 0.5125 \)
2. \( Q_2 = \frac{-97 - 94.95}{-4} = \frac{-191.95}{-4} = 47.9875 \)

Thus, the breakeven points occur when \( Q = 0.5125 \) and \( Q = 47.9875 \).

---

### Summary of Answers:
- **a. Total Revenue Function**: \( TR(Q) = 100Q - 2Q^2 \)
- **b. Profit Function**: \( \pi(Q) = 97Q - 2Q^2 - 50 \)
- **c. Breakeven Points**: \( Q = 0.51 \) and \( Q = 47.99 \)

---

Would you like further details or clarifications on any part of the solution?

Here are 5 questions you could explore further:
1. What happens to profit if the production quantity increases beyond the breakeven points?
2. How would a change in fixed costs affect the breakeven points?
3. What is the marginal revenue and marginal cost based on these functions?
4. How can we interpret the breakeven quantities in a real-world context?
5. What impact would price elasticity of demand have on this company's revenue function?

**Tip:** In optimization problems, finding the breakeven points is key to understanding the minimum and maximum levels of production for profitability.

Sebuah perusahaan memiliki fungsi biaya total TC = 50 + 3Q dan fungsi permintaan P = 100 - 2Q. a. Tentukan fungsi penerimaan total. b. Tentukan fungsi keuntungan. c. Hitunglah titik breakeven.

Solve a company's cost and demand function problem involving total revenue, profit function, and breakeven points. The solution involves key concepts like quadratic equations, breakeven analysis, and the quadratic formula. Ideal for university-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation