Download Lesson 4.2b

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)  7 x  5
b. f ( x)  x 2  10 x  20
c. f ( x)  6 x 2  24 x  20
3. The total revenue for Jane's Vacation Rentals is given as the function R( x)  200 x  0.4 x 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  40 x  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  0.5 x 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:
h(t )  16t 2  v0 t  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.