Modeling with Quadratic Functions

Each of the problems below is a link. If you click on the problem, it will take you to a YouTube video of a full, step-by-step solution for the problem. You may try these problems as practice and use the videos to check your answer, or you can just watch the videos to see more worked-out example problems.

The gas mileage M for a certain car is modeled by the function M of s = negative 1 twenty eighth s squared + 3s minus 31 for 15 is less than or equal to s, which is less than or equal to 70, where s is the speed in miles per hour and M is measured in miles per gallon. What is the car’s best gas mileage, and at what speed is it attained?
A basketball team plays in an arena that has a seating capacity of 15,000 spectators. With the ticket price set at $14, average attendance at recent games has been 9500. A market survey indicates that for each dollar the ticket price is lowered, the average attendance increases by 1000. Find a function that models the total revenue in terms of ticket price. What is the price that maximizes revenue? What is the corresponding attendance?
A farmer has 120 yards of fencing to enclose three sides of a rectangular patch of land, which is bordered by a river on the fourth side. Write a function that models the area enclosed with respect to the side length x perpendicular to the river. What dimensions would produce the maximum area? How much area can he enclose?