Questions: It has been determined that the cost of producing x units of a certain item is 4x+1275. The demand function is given by p=D(x)=42-0.2x. Find the profit function.

It has been determined that the cost of producing x units of a certain item is 4x+1275. The demand function is given by p=D(x)=42-0.2x. Find the profit function.

Transcript text: Correct It has been determined that the cost of producing $x$ units of a certain item is $4 x+1275$. The demand function is given by $p=D(x)=42-0.2 x$. Step 2 of 2: Find the profit function.

Solution

Solution Steps

Step 1: Calculate the Revenue Function

The revenue function, $R(x)$, is calculated as $R(x) = x(c - dx) = 42x - 0.2x^2$.

Step 2: Calculate the Cost Function

The cost function, $C(x)$, is given by $C(x) = ax + b = 4x + 1275$.

Step 3: Derive the Profit Function

The profit function, $P(x)$, is the difference between the revenue and cost functions: $P(x) = R(x) - C(x) = (42x - 0.2x^2) - (4x + 1275) = -0.2x^2 + (42 - 4)x - 1275$.

Step 4: Simplify the Profit Function

The simplified profit function is $P(x) = -0.2x^2 + (42 - 4)x - 1275$, and for $x = 1$, the profit is approximately -1237.2.