Given demand function of a product:

P = (1154/2) - (7/2)Q

The firm has a fixed cost of $418 and unit cost of $21.2.

a. What is the quantity when total revenue is maximum?

(use 3 digits after decimal)

To find the quantity when total revenue is maximized, we first need to derive the total revenue function and then determine the quantity at which this revenue is maximized.

### Step 1: Express the Total Revenue Function
The demand function is given by:

\[
P = \frac{1154}{2} - \frac{7}{2}Q = 577 - 3.5Q
\]

Total Revenue (TR) is given by:

\[
TR = P \times Q
\]

Substituting the demand function into the total revenue equation:

\[
TR = (577 - 3.5Q) \times Q
\]

This simplifies to:

\[
TR = 577Q - 3.5Q^2
\]

### Step 2: Differentiate the Total Revenue Function
To find the quantity that maximizes total revenue, we need to take the derivative of the total revenue function with respect to $Q$ and set it equal to zero:

\[
\frac{d(TR)}{dQ} = 577 - 7Q
\]

### Step 3: Set the Derivative Equal to Zero and Solve for $Q$

\[
577 - 7Q = 0
\]

\[
7Q = 577
\]

\[
Q = \frac{577}{7} \approx 82.429
\]

So, the quantity that maximizes total revenue is **82.429 units**.

Would you like further details or have any other questions?

Here are 5 relative questions to expand on the topic:

1. How do fixed and variable costs affect the profit function?
2. What is the relationship between marginal revenue and marginal cost at the profit-maximizing quantity?
3. How would a change in the unit cost affect the profit-maximizing quantity?
4. How do you calculate the break-even quantity for a firm?
5. What is the significance of elasticity of demand in revenue maximization?

**Tip:** The quantity that maximizes total revenue occurs where marginal revenue equals zero.

Given demand function of a product: P = (1154/2) - (7/2)Q The firm has a fixed cost of $418 and unit cost of $21.2. a. What is the quantity when total revenue is maximum? (use 3 digits after decimal)

Learn how to determine the quantity that maximizes total revenue for a given demand function using calculus. This problem involves key concepts such as algebra, differentiation, and revenue maximization, making it suitable for students in Grades 11-12.

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