This is the example of a problem and is solved.:

Raggs, Lid. a clothing firm, determines that in order to sell x suits, the price per suit must be p = 150 - 0.5x. It also determines that the total cost of producing x suits is given by C(x) = 4000 + 0.25x-. a) Find the total revenue, R(x). b) Find the total profit, P(x). c) How many suits must the company produce and sell in order to maximize profit? d) What is the maximum profit? e) What price per suit must be charged in order to maximize profit?A university is trying to determine what price to charge for tickets to football games. At a price of $27 per ticket, attendance averages 40,000 people per game. Every decrease of $3 adds 10,000 people to the average number. Every person at the game spends an average of $3.00 on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price? Make a list of the variables and constants in the problem. Let x represent the decrease in ticket price from $27. If the ticket price increases, then x is negative. Write the attendance, price per ticket, and revenue in terms of this variable. Start with the average attendance. Notice that the ratio of the change in attendance to the change in price is a constant, which means the slope of the equation must be a constant. Therefore, the attendance must have the form y = mx + b. The attendance increases by 10,000 for a decrease of $3 in the ticket price. Also, when the ticket price is $27, then the attendance is 40,000. Use this information to write a linear equation for the attendance in terms of x. 10.000 y = 40,000 + 3 XA university is trying to determine what price to charge for tickets to football games. At a price of $27 per ticket, attendance averages 40,000 people per game. Every decrease of $3 adds 10,000 people to the average number. Every person at the game spends an average of $3.00 on concessions. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price? Make a list of the variables and constants in the problem. Let x represent the decrease in ticket price from $27. If the ticket price increases, then x is negative. Write the attendance, price per ticket, and revenue in terms of this variable. Start with the average attendance. Notice that the ratio of the change in attendance to the change in price is a constant, which means the slope of the equation must be a constant. Therefore, the attendance must have the form y = mx + b. The attendance increases by 10,000 for a decrease of $3 in the ticket price. Also, when the ticket price is $27, then the attendance is 40,000. Use this information to write a linear equation for the attendance in terms of x. 10.000 y = 40,000 + 3 XWrite the price per ticket, p, in terms of x as well. p = 27-x The revenue has two parts, the price per ticket multiplied by the number of people and the averag concession spending multiplied by the number of people. 10,000 Use y = 40,000 + 3 x and p =27 - x to write the revenue function, R(x). R(x) = (y)(p) + (y)(3.00) 10,000 10,000 R(x) = 40,000 + 3 x (27 - x) + 40,000 + 3 x (3.00) R(x) = 10000 2 3 + 60,000x + 1,200,000From the problem statement, the revenue is the quantity to be maximized. 10000 The value to be maximized is R(x) = - - 3 - +60.000x + 1,200,000, and is already written in terms of a single variable. To maximize the function, first find the derivative R'(x). 20,000 R'(X) = - - 3 x + 60,000 Since R'(x) is defined for all x, the only critical values occur where R'(x) =0. Solve this equation. 20,000 x + 60,000 = 0 3 x = 9.00 There is only one critical value. Use the second derivative, R"(x), to determine if it is a maximum orminimum. 20,000 R'(x) = - 3 x + 60,000 20000 R"(X) = - 3 Since the second derivative is negative at x =9.00, the critical point is a maximum. Find the new ticket price using the equation found previously. p = 27-x p = 18.00 Find the attendance using the corresponding equation.10,000 y = 40.000 + 3 X y = 70.000 Therefore, the revenue is maximized when the price per ticket is $18.00. At that price, the average attendance will be 70,000 people