A company uses the function `P (x) =150x - x ^(2)` to determine how much profit the company will make when it sells 150 uinits of a certain product that sells for x dollars per unit. How much more profit per unit, in dollars, will the company make if it charges `$25` for the product than if it charges `$20` ? (Ignore the dollar sign when gridding your response.)

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the Profit Function
The profit function is given by:
\[ P(x) = 150x - x^2 \]
where \( x \) is the selling price per unit.

### Step 2: Calculate Profit at $25
We need to find the profit when the selling price is $25:
\[ P(25) = 150(25) - (25)^2 \]
Calculating this:
\[ P(25) = 3750 - 625 = 3125 \]

### Step 3: Calculate Profit at $20
Next, we find the profit when the selling price is $20:
\[ P(20) = 150(20) - (20)^2 \]
Calculating this:
\[ P(20) = 3000 - 400 = 2600 \]

### Step 4: Find the Difference in Profit
Now, we calculate the difference in profit between selling at $25 and $20:
\[ \text{Difference} = P(25) - P(20) \]
\[ \text{Difference} = 3125 - 2600 = 525 \]

### Step 5: Calculate Profit Per Unit
To find the profit per unit, we divide the difference in profit by the number of units sold (150):
\[ \text{Profit per unit} = \frac{525}{150} \]
Calculating this:
\[ \text{Profit per unit} = 3.5 \]

### Final Answer
Thus, the company will make an additional profit of **3.5** dollars per unit when it charges $25 instead of $20.

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