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 that maximizes total revenue, we can follow these steps:

### 1. **Express the Total Revenue Function:**

The total revenue (TR) is given by the product of the price (P) and quantity (Q):

\[
TR = P \times Q
\]

Given the demand function:

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

So, the total revenue function becomes:

\[
TR = Q \times (577 - 3.5Q) = 577Q - 3.5Q^2
\]

### 2. **Find the Quantity that Maximizes Total Revenue:**

To maximize the total revenue, take the derivative of the $ TR $ function with respect to $ Q $ and set it equal to zero:

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

Set the derivative equal to zero to find the critical point:

\[
577 - 7Q = 0
\]

\[
7Q = 577
\]

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

So, the quantity $ Q $ that maximizes total revenue is **82.429** (rounded to three decimal places).

Would you like further details or explanations?

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Here are some related questions:
1. What is the maximum total revenue?
2. How do we calculate the profit at the revenue-maximizing quantity?
3. What is the price at the revenue-maximizing quantity?
4. How would fixed costs impact profit at this quantity?
5. What is the elasticity of demand at the revenue-maximizing quantity?

**Tip:** The revenue-maximizing quantity occurs where the marginal revenue equals zero, not necessarily where profit is maximized.

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 maximize total revenue from the demand function P = 577 - 3.5Q. This problem involves calculus concepts like differentiation and revenue functions. We walk you through step-by-step calculations to find the revenue-maximizing quantity, with answers rounded to three decimal places.

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