The total revenue received from the sale of x units of a product is given by `R(x) = 20 x - 0.5x^(2)`. Find
Difference of average revenue and marginal revenue when x = 10

Answer

To solve the problem, we need to find the difference between the average revenue and marginal revenue when \( x = 10 \).

### Step 1: Define the Total Revenue Function
The total revenue \( R(x) \) is given by:
\[ R(x) = 20x - 0.5x^2 \]

### Step 2: Calculate Average Revenue
Average revenue \( AR \) is defined as total revenue divided by the number of units sold:
\[ AR(x) = \frac{R(x)}{x} = \frac{20x - 0.5x^2}{x} \]
Simplifying this gives:
\[ AR(x) = 20 - 0.5x \]

### Step 3: Calculate Marginal Revenue
Marginal revenue \( MR \) is the derivative of the total revenue function with respect to \( x \):
\[ MR(x) = \frac{dR}{dx} = \frac{d}{dx}(20x - 0.5x^2) \]
Using the power rule for differentiation:
\[ MR(x) = 20 - x \]

### Step 4: Evaluate Average Revenue and Marginal Revenue at \( x = 10 \)
Now we will substitute \( x = 10 \) into the average revenue and marginal revenue formulas.

**Average Revenue at \( x = 10 \)**:
\[ AR(10) = 20 - 0.5(10) = 20 - 5 = 15 \]

**Marginal Revenue at \( x = 10 \)**:
\[ MR(10) = 20 - 10 = 10 \]

### Step 5: Find the Difference
Now, we calculate the difference between average revenue and marginal revenue:
\[ \text{Difference} = AR(10) - MR(10) = 15 - 10 = 5 \]

### Final Answer
The difference between average revenue and marginal revenue when \( x = 10 \) is:
\[ \boxed{5} \]