Analysis Name___________________
Quadratics III Date____________________
1. Find two positive real numbers whose product is a maximum if the sum of the
first and twice the second is 24.
x 6, y 12
2. Let x y 16 . Find the minimum value of the quantity x 2 y 2.
x 8, y 8
3. Find two positive real numbers whose product is a maximum if the sum of the
first and three times the second is 42.
x 21, y 7
1 2
4. The height y (in feet) of a ball thrown by a child is y x 2x 4 , where x
12
is the horizontal distance (in feet) from the point at which the ball is thrown.
(Use your graphing calculator for this problem)
a. What is the maximum height of the ball?
16 feet
b. How far from the child will the ball strike the ground?
25.86 feet
c. What is the initial height of the ball?
4 feet
5. The Lower Moreland Student Store is selling "Official Mr. Cendrowski Lunch
Boxes" at a price of $18 each. Each lunch box comes with a thermos and surprise
math problem inside of each one. At this price they can sell 180 lunch boxes per
month. For each $1 increase in the selling price, the store sells 12 fewer lunch
boxes per month. Each lunch box has a production cost of $2.50. What is the
optimum price at which to sell their lunchboxes?
$16.50
Analysis Page 2
Quadratics II
6. A rancher wants to build a corral according to the diagram shown. He has a total
of 2000 feet of fencing. Find the maximum area of the corral.
500 feet x 200 feet
7. A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet
per second and at an angle of 45° with respect to the ground. The path of the
baseball is given by the function
f (x ) .0032x 2 x 3 ,
where f(x) is the height of the baseball (in feet) and x is the horizontal distance
from home plate (in feet). What is the maximum height reached by the baseball?
How far from home plate did the baseball land? (This is a calculator problem)
Max height 156.25 feet
Distance 315.47 feet
8. Mr. C has gone into table building business and has hired master table builder
Kent to help him run the company. Mr. C’s deep mathematical insight told him that
each of the 30 people under Kent are averaging sales of $5,000 per month. Mr. C’s
research also told him that for every additional person Mr. C hired, the average
sale would decrease by $100 per person. How many people should Mr. C hire to
maximize profits?
10 people
Mr. John Cendrowski
Analysis
Analysis Page 3
Quadratics II
9. A major snowstorm is headed for most of Pennsylvania. The National Weather
Service has determined that the path of a storm is parabolic. If the storm passes
through map coordinates (4,25), (2,9), and (-2, -11), find the rule that represents
the general path of the storm.
f (x ) .5x 2 5x 3
10. A doorway has the shape of a parabolic arch and is 9 feet high at the center
and 6 feet wide at the base. If rectangular box 8 feet high must fit through the
doorway, what is the maximum width the box can have?
2 feet
11. The publisher of a magazine that has a circulation of 80,000 and sells for $1.60
per copy decides to raise the price of the magazine because of production and
distribution costs. By surveying the readers, the publisher learns that for each
40-cent increase in the price of the magazine, he will lose 10,000 readers. What
should the price per copy be to maximize the income?
$2.40
12. Find the number of units sold that yields a maximum annual revenue for a
company that produces health food supplements. The total revenue (in dollars) is
given by R 900x .1x 2 , where x is the number of units sold.
4500 units
13. A manufacturer of lighting fixtures has daily production costs of
C 800 10x .25x 2 , where C is the total cost (in dollars) and x is the number of
units produced. How many fixtures should be produced per day to yield a minimum
cost?
20 units
Mr. John Cendrowski
Analysis
Analysis Page 4
Quadratics II
14. Manufacturer finds that the revenue generated by selling x units of a certain
commodity is given by the function R (x ) 80x .4x 2 , where R(x) is measured in
dollars. What is the maximum revenue, and how many units should be
manufactured to obtain this maximum?
Maximum Revenue 4000
100 units
15. Mr. C is throwing a private party on the grounds of the high school. The
district has given Mr. C 240 feet of fence in which to construct an area for his
guests. If Mr. C desires to construct a rectangular area for his guests along with a
dividing fence to separate his appealing from his less appealing guests, what is the
maximum area that Mr. C could have for his party. (Assume that the dividing fence
is parallel to one of the 4 sides of the rectangle.)
60 by 40 feet
16. How far from the origin is the vertex of the parabola y x 2 6x 13 ?
5
17. Let R .4x 2 10x 5 and C .5x 2 2x 101 . For which value of x is R C
a maximum?
49
18. Find the distance between the vertices of the parabolas y x 2 4x 6 and
y x 2 4x 5 .
5
19. How far from the origin is the vertex of the parabola y x 2 6x 13 ?
5
20. For which value of c will the minimum value of the function f (x ) x 2 2x c
be at 2 ? (The minimum value of a quadratic is the y value of the vertex)
c 2 1
Mr. John Cendrowski
Analysis
Analysis Page 5
Quadratics II
21. Find the equation of the line that passes through the vertices of the two
1 81
parabolas f (x ) x 2 4x 1 and f (x ) x 2 9x .
2 2
130
Mr. John Cendrowski
Analysis