Questions: Suppose the price-demand and cost functions for the production of cordless drills is given respectively by p=0.03 x+15 and C(x)=89 x+5587, where x is the number of cordless drills that are sold at a price of p dollars per drill and C(x) is the cost of producing x cordless drills. Find the profit function, P(x).

Suppose the price-demand and cost functions for the production of cordless drills is given respectively by p=0.03 x+15 and C(x)=89 x+5587, where x is the number of cordless drills that are sold at a price of p dollars per drill and C(x) is the cost of producing x cordless drills.

Find the profit function, P(x).

Transcript text: Question Suppose the price-demand and cost functions for the production of cordless drills is given respectively by $p=0.03 x+15$ and $C(x)=89 x+5587$, where $x$ is the number of cordless drills that are sold at a price of $p$ dollars per drill and $C(x)$ is the cost of producing $x$ cordless drills. Find the profit function, $P(x)$.

Solution

Solution Steps

To find the profit function $ P(x) $, we need to determine the revenue function $ R(x) $ and then subtract the cost function $ C(x) $ from it. The revenue function is given by the product of the price per unit $ p $ and the number of units sold $ x $. Once we have $ R(x) $, we can calculate the profit function as $ P(x) = R(x) - C(x) $.

Step 1: Determine the Revenue Function

The revenue function $ R(x) $ is calculated by multiplying the price-demand function $ p(x) = 0.03x + 15 $ by the number of units sold $ x $. Thus, the revenue function is: \[ R(x) = x \cdot (0.03x + 15) = 0.03x^2 + 15x \]

Step 2: Determine the Profit Function

The profit function $ P(x) $ is the difference between the revenue function $ R(x) $ and the cost function $ C(x) = 89x + 5587 $. Therefore, the profit function is: \[ P(x) = R(x) - C(x) = (0.03x^2 + 15x) - (89x + 5587) \] Simplifying, we get: \[ P(x) = 0.03x^2 + 15x - 89x - 5587 = 0.03x^2 - 74x - 5587 \]

Step 3: Calculate the Profit for $ x = 100 $

Substitute $ x = 100 $ into the profit function: \[ P(100) = 0.03(100)^2 - 74(100) - 5587 \] \[ P(100) = 0.03 \times 10000 - 7400 - 5587 \] \[ P(100) = 300 - 7400 - 5587 \] \[ P(100) = -12687 \]