1.) The price of a certain product varies according to how much is available. At a production level of x units the price per units the price per unit is 3.40-0.01x.
a.) What is the revenue if x units are produced and sold?
b.) If the cost of producing x units is 100+1.20x, find the profit function and the break-even points.
2.) A theater company charges the same price for all tickets, regardless of day of the week or seat location. It has found that a price of $16 it sells 900 tickets a week, but at a price of $20 per ticket it only sells 780. Assume that q, the number of tickets sold, is a linear function of p.
a.) Write an equation expressing q as a function of p (demand curve).
b.) Write the revenue as a function of the price.
3.) A restaurant has determined that its monthly cost function is given by C=3,000+16q, where q is the number of dinners sold per month. Moreover, q=1,200-20p, where p is the average price of a dinner.
a.) Solve for p in terms of q and write the revenue and profit as functions of q.
b.) Find the break-even points in terms of q. In other words, find the minimum and maximum number of dinners per month that the restaurant can serve without losing money.
c.) Economists usually take the price as the independent variable because it can be controlled by the company. Write the cost, revenue and profit functions of p and find the break-even points in terms of p.
d.) Determine the range of prices for which the restaurant makes a profit.
4.) In exercise 22, what price should the restaurant charge to maximize:
a.) its revenue?
b.) its profit?
5.) A software company is currently selling an average of 400 copies of a certain computer game per week at a price of $120. It assumes the demand is a linear function of the price, and it estimates that every $5 decrease in price will add an average of 50 sales per week.
a.) Write the demand q as a linear function of the price p.
b.) Write the revenue as a function of the price.
6.) In exercise 24, what price should the software company charge to maximize its revenue from the sale of the computer game?