A product can be manufactured at a total cost `C(x)=x^2/(100) x+100x+40` , where x is the number of units produced. The price at which each unit can be sold is given by `P=(200-x/(400))`. Determine the production level x at which the profit is maximum. What is the price per unit and total profit at the level of production?

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define the Cost Function and Price Function
The total cost function is given by:
\[ C(x) = \frac{x^2}{100} + 100x + 40 \]

The price per unit is given by:
\[ P(x) = 200 - \frac{x}{400} \]

### Step 2: Calculate Total Selling Price
The total selling price (SP) for \( x \) units is:
\[ \text{Total SP} = x \cdot P(x) = x \left( 200 - \frac{x}{400} \right) \]
\[ \text{Total SP} = 200x - \frac{x^2}{400} \]

### Step 3: Define the Profit Function
Profit is defined as the total selling price minus the total cost:
\[ \text{Profit} = \text{Total SP} - C(x) \]
Substituting the expressions we have:
\[ P(x) = \left( 200x - \frac{x^2}{400} \right) - \left( \frac{x^2}{100} + 100x + 40 \right) \]

### Step 4: Simplify the Profit Function
Now, we simplify the profit function:
\[ P(x) = 200x - \frac{x^2}{400} - \frac{x^2}{100} - 100x - 40 \]
To combine the terms, we need a common denominator for the quadratic terms. The common denominator for 400 and 100 is 400:
\[ P(x) = 200x - 100x - 40 - \left( \frac{x^2}{400} + \frac{4x^2}{400} \right) \]
\[ P(x) = 100x - 40 - \frac{5x^2}{400} \]
\[ P(x) = 100x - 40 - \frac{x^2}{80} \]

### Step 5: Differentiate the Profit Function
To find the production level \( x \) at which profit is maximum, we differentiate \( P(x) \) with respect to \( x \):
\[ \frac{dP}{dx} = 100 - \frac{2x}{80} \]
Setting the derivative to zero to find critical points:
\[ 100 - \frac{x}{40} = 0 \]
\[ \frac{x}{40} = 100 \implies x = 4000 \]

### Step 6: Check for Maximum Profit
To confirm that this critical point is a maximum, we differentiate again:
\[ \frac{d^2P}{dx^2} = -\frac{1}{40} \]
Since \( \frac{d^2P}{dx^2} < 0 \), this indicates that the profit is maximized at \( x = 4000 \).

### Step 7: Calculate Price per Unit
Now, we calculate the price per unit at \( x = 4000 \):
\[ P(4000) = 200 - \frac{4000}{400} = 200 - 10 = 190 \]

### Step 8: Calculate Total Profit
Finally, we calculate the total profit at this production level:
\[ \text{Total Profit} = P(4000) = 100(4000) - \frac{(4000)^2}{80} - 40 \]
Calculating:
\[ = 400000 - \frac{16000000}{80} - 40 \]
\[ = 400000 - 200000 - 40 = 199960 \]

### Final Results
- The production level \( x \) at which the profit is maximum is \( 4000 \).
- The price per unit at this level is \( 190 \).
- The total profit at this level of production is \( 199960 \).