Question

A one-product company finds that its profit, P. in millions of dollars, is given by the following equation where a is the amount spent on advertising, in millions of dollars, and p is the price charged per item of the product, in dollars.

P(a.p)=2ap+80p-15p²-1/10a²p-120
Find the maximum value of P and the values of a and p at which it is attained.

Answer 1

To find the maximum value of the profit function P(a,p) = 2ap + 80p - 15p² - 1/10a²p - 120, take the partial derivatives with respect to a and p, set them equal to zero, solve the system of equations, and evaluate P at the critical points.

Step-by-step explanation:

The profit function, P(a,p), for the one-product company is given by P(a,p) = 2ap + 80p - 15p² - 1/10a²p - 120. To find the maximum value of P, we need to find the critical points of the function by taking the partial derivatives with respect to a and p and setting them equal to zero. Once we have the critical points, we can evaluate P at these points to find the maximum value.

To find the critical points:

1. Find the partial derivative of P(a,p) with respect to a and set it equal to zero: ∂P/∂a = 2p - (1/5)a²p = 0
2. Find the partial derivative of P(a,p) with respect to p and set it equal to zero: ∂P/∂p = 2a + 80 - 30p - (1/10)a² = 0
3. Solve the system of equations to find the values of a and p at the critical points.

Once we have the critical points, we can plug them into the profit function to find the maximum value of P.