To find the marginal revenue when 100 items are sold, follow these steps:
### Step 1: Identify the price function
The price function, [tex]\( p(x) \)[/tex], is given as:
[tex]\[ p(x) = 36 - 0.03x \][/tex]
### Step 2: Set up the revenue function
Revenue, [tex]\( R(x) \)[/tex], is the product of the price per item and the number of items sold:
[tex]\[ R(x) = x \cdot p(x) \][/tex]
Substitute [tex]\( p(x) \)[/tex] in the equation:
[tex]\[ R(x) = x \cdot (36 - 0.03x) \][/tex]
[tex]\[ R(x) = 36x - 0.03x^2 \][/tex]
### Step 3: Differentiate the revenue function to find the marginal revenue
The marginal revenue is the derivative of the revenue function with respect to [tex]\( x \)[/tex]:
[tex]\[ MR = \frac{dR}{dx} = \frac{d}{dx}(36x - 0.03x^2) \][/tex]
[tex]\[ MR = 36 - 0.06x \][/tex]
### Step 4: Evaluate the marginal revenue at 100 items
To find the marginal revenue when 100 items are sold, substitute [tex]\( x = 100 \)[/tex] into the marginal revenue function:
[tex]\[ MR = 36 - 0.06 \cdot 100 \][/tex]
[tex]\[ MR = 36 - 6 \][/tex]
[tex]\[ MR = 30 \][/tex]
### Step 5: Round the answer to two decimal places
Since the marginal revenue at 100 items is already a whole number, no rounding is needed. The answer is:
[tex]\[ 30.00 \][/tex]
This is the marginal revenue when 100 items are sold.
