Answer:
The profit function P(x) is defined as the difference between the revenue function R(x) and the cost function C(x): P(x) = R(x) - C(x). Substituting the given functions for R(x) and C(x), we get:
P(x) = (-31.72x^2 + 2030x) - (-126.96x + 26391) = -31.72x^2 + 2156.96x - 26391
To find the maximum profit, we need to find the vertex of this quadratic function. The x-coordinate of the vertex is given by the formula x = -b/(2a), where a = -31.72 and b = 2156.96. Substituting these values into the formula, we get:
x = -2156.96/(2 * (-31.72)) ≈ 34
Substituting this value of x into the profit function, we find that the maximum profit is:
P(34) = -31.72(34)^2 + 2156.96(34) - 26391 ≈ $4,665
The selling price of the dog food is given by the revenue function divided by x: R(x)/x = (-31.72x^2 + 2030x)/x = -31.72x + 2030. Substituting x = 34 into this equation, we find that the selling price of the dog food should be set to:
-31.72(34) + 2030 ≈ $92
So, the maximum profit of $4,665 can be made when the selling price of the dog food is set to $92 per bag.