TR = 5.000Q - 10Q² TC = 5Q²-10.000 Ditanya Q=.....? Profit maks?

To solve for the quantity \( Q \) that maximizes profit, we need to use the given equations for Total Revenue (TR) and Total Cost (TC), then derive the Profit function \( \pi(Q) \).

### Step 1: Define the profit function
The profit function is defined as:
\[
\pi(Q) = TR - TC
\]

Given:
\[
TR = 5,000Q - 10Q^2
\]
\[
TC = 5Q^2 - 10,000
\]

Substitute \( TR \) and \( TC \):
\[
\pi(Q) = (5,000Q - 10Q^2) - (5Q^2 - 10,000)
\]
\[
\pi(Q) = 5,000Q - 10Q^2 - 5Q^2 + 10,000
\]
\[
\pi(Q) = 5,000Q - 15Q^2 + 10,000
\]

### Step 2: Find the critical point
To find the quantity \( Q \) that maximizes profit, take the derivative of \( \pi(Q) \) with respect to \( Q \), and set it equal to zero:
\[
\frac{d\pi(Q)}{dQ} = 5,000 - 30Q = 0
\]

Solve for \( Q \):
\[
30Q = 5,000
\]
\[
Q = \frac{5,000}{30} = 166.67
\]

### Step 3: Determine the maximum profit
Now substitute \( Q = 166.67 \) back into the profit function to find the maximum profit:
\[
\pi(166.67) = 5,000(166.67) - 15(166.67)^2 + 10,000
\]
Let's compute this:

\[
\pi(166.67) = 833,350 - 15 \cdot 27,777.78 + 10,000
\]
\[
\pi(166.67) = 833,350 - 416,666.7 + 10,000
\]
\[
\pi(166.67) = 426,683.3
\]

### Final Answer
- The quantity \( Q \) that maximizes profit is **166.67**.
- The maximum profit is approximately **426,683.3**.

Would you like a more detailed explanation or have any questions?

### 5 Related Questions:
1. How do you determine the derivative of a profit function?
2. What is the significance of the second derivative in finding maxima or minima?
3. How do fixed costs affect the profit-maximizing quantity?
4. Can this problem be solved using a graphical approach?
5. How would the profit-maximizing \( Q \) change if TC was different?

### Tip:
Remember that when maximizing or minimizing a function, the first derivative helps find critical points, while the second derivative test confirms whether it's a maximum or minimum!

TR = 5,000Q - 10Q² TC = 5Q² - 10,000. Ditanya Q=.....? Profit maks?

Learn how to maximize profit for the functions TR = 5,000Q - 10Q² and TC = 5Q² - 10,000 using derivatives. This problem demonstrates the use of the profit function and first derivative test to find the optimal quantity Q and maximum profit, with step-by-step calculations. Suitable for Grades 10-12.

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