A TV manufacturer has the total cost function C(x)=72x+3600 (in dollars). The manufacturer sells the TVs to consumers for $372 each.  Let x represent the number of TVs manufactured and sold. 

What is the manufacturer's revenue function, R(x)?   R(x)=

What is the manufacturer's simplified profit function, P(x)?   P(x)=

What is the revenue from selling 336 items?   $

What is the cost of producing 336 items?   $  

What is the profit on 336 items?   $

Let's work through each part of the problem step by step.

### 1. **Revenue Function**:
The revenue function $ R(x) $ represents the total money earned from selling $ x $ items. Since each TV is sold for $372, the revenue function is:

\[
R(x) = 372x
\]

---

### 2. **Profit Function**:
The profit function $ P(x) $ is given by the difference between revenue and cost, i.e.,

\[
P(x) = R(x) - C(x)
\]

Substitute $ R(x) = 372x $ and $ C(x) = 72x + 3600 $ into the equation:

\[
P(x) = 372x - (72x + 3600)
\]

Simplify the equation:

\[
P(x) = 372x - 72x - 3600
\]
\[
P(x) = 300x - 3600
\]

---

### 3. **Revenue from Selling 336 Items**:
To find the revenue from selling 336 items, substitute $ x = 336 $ into the revenue function:

\[
R(336) = 372 \times 336
\]

Calculating:

\[
R(336) = 125,092
\]

So, the revenue from selling 336 items is **$125,092**.

---

### 4. **Cost of Producing 336 Items**:
To find the cost of producing 336 items, substitute $ x = 336 $ into the cost function:

\[
C(336) = 72 \times 336 + 3600
\]

Calculating:

\[
C(336) = 24,192 + 3600 = 27,792
\]

So, the cost of producing 336 items is **$27,792**.

---

### 5. **Profit on 336 Items**:
To find the profit from selling 336 items, substitute $ x = 336 $ into the profit function:

\[
P(336) = 300 \times 336 - 3600
\]

Calculating:

\[
P(336) = 100,800 - 3600 = 97,200
\]

So, the profit from selling 336 items is **$97,200**.

---

### Summary of Results:
1. **Revenue function**: $ R(x) = 372x $
2. **Profit function**: $ P(x) = 300x - 3600 $
3. **Revenue from selling 336 items**: **$125,092**
4. **Cost of producing 336 items**: **$27,792**
5. **Profit on 336 items**: **$97,200**

Would you like more details or have any questions?

Here are five related questions:
1. What would the profit be if the manufacturer produced 500 items?
2. How would the profit change if the price per TV were increased to $400?
3. What is the break-even point (number of items sold where profit is zero)?
4. How does a change in fixed costs affect the profit function?
5. What is the marginal revenue for each additional TV sold?

**Tip:** When calculating profit, always consider both fixed and variable costs. This helps in understanding how scaling production affects profitability.

A TV manufacturer has the total cost function C(x) = 72x + 3600 (in dollars). The manufacturer sells the TVs to consumers for $372 each. Let x represent the number of TVs manufactured and sold. What is the manufacturer's revenue function, R(x)? What is the manufacturer's simplified profit function, P(x)? What is the revenue from selling 336 items? What is the cost of producing 336 items? What is the profit on 336 items?

Learn how to calculate the revenue, cost, and profit for a TV manufacturer given a linear cost function C(x) = 72x + 3600 and revenue function R(x) = 372x. Includes a step-by-step solution for determining the profit on 336 items. Suitable for Grades 9-11.

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