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A home business examines its monthly production costs and sales and finds the cost function when [tex][tex]$x$[/tex][/tex] items are produced is given by:
[tex]\[ C(x) = 1419 + 24x - 0.056x^2 \text{ (in dollars) } \][/tex]
and the price each item can be sold for is given by:
[tex]\[ p(x) = 36 - 0.03x \text{ (in dollars) } \][/tex]

Find the marginal revenue if 100 items are sold. Round your answer to two decimal places and do not include units.

Sagot :

GinnyAnswer
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.
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