The marginal profit equation for the company can be represented as follows

The Case

In a supposed scenario, let's consider a company that manufactures shampoo by employing AI  technology to analyze consumer demand information. The analysis of the consumer model reveals an interesting finding: an inverse relationship exists between the price of shampoo and its marginal profit. Armed with this knowledge, the firm strategically decides to produce the optimal quantity of shampoo that maximizes its profit. The marginal profit equation for the company can be represented as follows:

dπ/dQ = π’(Q)= 300-30Q

Requirement:

1. Calculate the general equation of profit from the given marginal profit equation of the Shampoo. 

2. If the profit of selling 5 units of a product equals $250, what is the definite solution?

(Q=5) = π(5) = 250

Solution:

To calculate the general equation of profit from the given marginal profit equation, we can integrate the marginal profit equation with respect to Q (quantity) to obtain the profit equation.

Given:

dπ/dQ = π'(Q) = 300 - 30Q

Integrating both sides with respect to Q:

∫ dπ = ∫ (300 - 30Q) dQ

Integration of dπ gives us the general profit equation:

π(Q) = ∫ (300 - 30Q) dQ

To find the definite solution when Q = 5, we can substitute the value of Q into the profit equation:

π(5) = ∫[300 - 30(5)] dQ

π(5) = ∫[300 - 150] dQ

π(5) = ∫150 dQ

π(5) = 150Q + C

Since we are given that the profit of selling 5 units is $250, we can use this information to solve for the constant C:

250 = 150(5) + C

250 = 750 + C

C = 250 - 750

C = -500

Therefore, the solid solution for the profit equation when Q = 5 is:

π(5) = 150Q - 500