Question

Given the demand function p=46−2q, find the total revenue function. What is the maximum total revenue?

Note: Enter your answer as a number only, without any special characters. For example, if you answer is $1,000, enter this as follows: 1000

Answer 1

The total revenue function is TR = 46q−2q^2. The maximum total revenue occurs at an output quantity of 11.5, and the maximum total revenue is 262.25.

Step-by-step explanation:

To find the total revenue function, we need to multiply the price (p) by the quantity (q). The demand function is given as p=46−2q. Therefore, the total revenue function (TR) is TR = p * q. Substituting the value of p from the demand function, we have TR = (46−2q) * q. Simplifying this equation, we get TR = 46q−2q^2.

To find the maximum total revenue, we need to find the value of q that maximizes TR. This can be done by finding the vertex of the quadratic equation. The vertex of a quadratic equation in the form ax^2 + bx + c is given by the x-coordinate x = -b / (2a). In this case, the vertex occurs at q = -46 / (2*-2) = 11.5. Therefore, the maximum total revenue occurs at an output quantity of 11.5.

To calculate the maximum total revenue, we substitute this value of q back into the total revenue function. TR = 46(11.5)−2(11.5)^2 = 529 - 266.75 = 262.25. Hence, the maximum total revenue is 262.25.