College Algebra: Lesson 4

College Algebra: Lesson 4.2B Max/Min Applications of Quadratic Functions

Recall: If a > 0, the parabola opens up. If a < 0, the parabola opens down.

Examples: 1. Among all rectangles that have a perimeter of 98, find the dimensions of the one whose area is the largest.

2. Determine whether the following function has a maximum, a minimum, or neither. If it has either a maximum or a minimum, find what that value is and where it occurs. a. f (x)  7x  5 b. f (x)  x 2 10x  20 c. f (x)  6x 2  24x  20

3. The total revenue for Jane's Vacation Rentals is given as the function R(x)  200x  0.4x 2 , where x is the number of villas filled. What number of villas filled produces the maximum revenue?

4. The total cost of producing a type of boat is given by C(x)  23000  40x  0.1x 2 , where x is the number of boats produced. How many boats should be produced to incur minimum cost? 5. The revenue for a bicycle shop is given by R(x)  x  p(x) dollars where x is the number of units sold and p(x)  400  0.5x is the unit price. Find the maximum revenue.

6. A projectile is launched upward with a velocity of 128 feet per second from the top of an 85-foot structure. What is the maximum height attained by the projectile? Height Function: 2 h(t)  16t  v0t  h0

7. A small cruising ship that can hold up to 72 people provides three-day excursions to groups of 44 or more. if the group contains 44 people, each person pays $58. The cost per person for all members of the party is reduced by $1 for each person in excess of 44. Find the size of the group that maximizes the income for the owners of the ship.

8. A rancher has 400 feet of fencing to put around a rectangular field and then subdivide the field into 2 identical smaller rectangular plots by placing a fence parallel to the field's shorter sides. Find the dimensions that maximize the enclosed area. 9. The back of George's property is a creek. George would like to enclose a rectangular area, using the creek as one side and fencing for the other three sides, to create a pasture. If there is 580 feet of fencing available, what is the maximum possible area of the pasture?

10. Among all pairs of numbers (x,y) such that 2x + y = 11, find the pair for which the sum of the squares, x 2  y 2 , is a minimum.