Questions: The linear profit function is P(x)=px-37x.

Transcript text: The linear profit function is $P(x)=p x-37 x$.

Solution

Solution Steps

To find the linear profit function $ P(x) $, we need to determine the profit per maintenance call. The profit function is generally given by the revenue function minus the cost function. Here, the revenue function is $ R(x) = 44x $ and the cost per call is given as $ 37x $. Therefore, the profit function can be expressed as the difference between these two functions.

Step 1: Define the Revenue and Cost Functions

The revenue function for $ x $ maintenance calls is given by: \[ R(x) = 44x \]

The cost function for $ x $ maintenance calls is given by: \[ C(x) = 37x \]

Step 2: Derive the Profit Function

The profit function $ P(x) $ is the difference between the revenue and the cost functions: \[ P(x) = R(x) - C(x) = 44x - 37x \]

Simplifying the expression, we get: \[ P(x) = 7x \]

Step 3: Calculate the Profit for 10 Maintenance Calls

Substitute $ x = 10 $ into the profit function: \[ P(10) = 7 \times 10 = 70 \]