MTH 105 A & C, Spring 2024 Maximizing Revenue & Profit: Problems

1.) Suppose that at a certain car dealership, the number of cars of a particular model that are sold through an average week is 25 when the price is set at $20,000. But for each $500 increase in price, the number of sales per week goes down by 1 car. How should the price of the car be set by the dealership in order to maximize revenue?

2.) At a particular college with a large football stadium, it has been true that the number of ticket sales for a home game averages 50,000 when ticket price is set at $20. But when the price is reduced to $16, the average attendance rises to 58,000. How should ticket prices be set by the college so that revenue is maximized? Assume a linear relationship between price and ticket sales.

3.) When a certain retailer sells a particular item for $120 each, the typical number of sales per week is 300 items. But for each $20 increase in price, the number of sales comes down by 15 per week. In order to maximize weekly revenue from sales of the item, how should its price be set?

4.) When a large concert hall sells tickets for $25 each, their typical sales for a show are 1000 tickets. But when the price of a ticket is reduced to $24, sales increase to 1040 tickets. Assuming a linear relationship between ticket price and sales, how should the concert hall set its price so that revenue from sales is maximized?

5.) A small retailer makes and sells a certain item. When they set their price at $50 per item, they average 290 sales per month. However, for each $2 increase in price, their number of sales drops by 10 per month. Also, assume that it costs the retailer $8 each to manufacture the item. How should they set the price of their item in order to maximize profit?

6.) At a roadside lemonade stand, the average number of cups sold on a hot summer day is 200 when the price per cup is $1. But when the price per cup is raised to $2, the average number of cups sold goes down to 150. Suppose that it costs the seller $0.50 of investment for each cup sold (for the actual paper cup and the lemonade as well). How should the price of a cup of lemonade be set so that profit is maximized? Assume a linear relationship between price and sales.