Suppose a company has fixed costs of $1800 and variable costs per unit of 15/16x + 1110 dollars, where x is the total number of units  produced. Suppose further that the selling price of its product is 1200- 1/16x dollars per unit. Find the maximum revenue.

The revenue $ R(x) $ for a company is calculated as the selling price per unit multiplied by the number of units sold. The problem gives the selling price per unit as:

\[
\text{Price per unit} = 1200 - \frac{1}{16}x
\]

Thus, the total revenue function is:

\[
R(x) = x \times \left(1200 - \frac{1}{16}x\right)
\]

Expanding this, we get:

\[
R(x) = 1200x - \frac{1}{16}x^2
\]

Now, to find the maximum revenue, we need to find the critical points by taking the derivative of $ R(x) $ and setting it equal to zero.

### Step 1: Differentiate the revenue function

\[
R'(x) = \frac{d}{dx} \left(1200x - \frac{1}{16}x^2\right)
\]

\[
R'(x) = 1200 - \frac{1}{8}x
\]

### Step 2: Set $ R'(x) = 0 $ to find the critical points

\[
1200 - \frac{1}{8}x = 0
\]

Solving for $ x $:

\[
\frac{1}{8}x = 1200
\]

\[
x = 1200 \times 8 = 9600
\]

### Step 3: Verify that this critical point is a maximum by using the second derivative test

The second derivative of $ R(x) $ is:

\[
R''(x) = \frac{d}{dx} \left(1200 - \frac{1}{8}x\right) = -\frac{1}{8}
\]

Since $ R''(x) = -\frac{1}{8} $ is negative, the function is concave down at $ x = 9600 $, indicating that this is a maximum point.

### Step 4: Calculate the maximum revenue

Substitute $ x = 9600 $ into the revenue function:

\[
R(9600) = 1200(9600) - \frac{1}{16}(9600^2)
\]

Now calculating:

\[
R(9600) = 1200 \times 9600 - \frac{1}{16} \times (9600)^2
\]

\[
R(9600) = 11520000 - \frac{1}{16} \times 92160000
\]

\[
R(9600) = 11520000 - 5760000
\]

\[
R(9600) = 5760000
\]

Thus, the maximum revenue is **$5,760,000**.

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### Related questions:
1. How do we calculate the total profit instead of just revenue?
2. What would be the effect on revenue if the selling price changed?
3. How does variable cost impact the maximum revenue?
4. Can we find the break-even point using this information?
5. What is the role of fixed costs in determining the overall profitability?

**Tip:** In problems involving revenue, profit, or cost functions, always identify key points (maximums/minimums) by differentiating and solving for critical points.

Suppose a company has fixed costs of $1800 and variable costs per unit of 15/16x + 1110 dollars, where x is the total number of units produced. Suppose further that the selling price of its product is 1200 - 1/16x dollars per unit. Find the maximum revenue.

Learn how to find the maximum revenue for a company with fixed and variable costs. This detailed solution covers key concepts such as revenue functions, critical points, and the application of derivatives. Suitable for college-level students studying calculus and economics.

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