Download Pre-Calculus - Applications of Quadratic Functions

Pre-Calculus Applications of Quadratic Functions
Name ____________________________
1. Trajectory of a ball. The height y (in feet) of a ball thrown by a child is: y  
1 2
x  2 x  4 ; where x is the
12
horizontal distance (in feet) from where the ball is thrown. (See figure).
a. How high was the ball when it left the child’s hand?
b. How high was the ball when it was at its maximum height?
c. How far from the child did the ball strike the ground?
2. Find two positive numbers whose sum is 110 and has a product that is a maximum.
3. The revenue R for a symphony concert is given by the equation R  
1
( x 2  4800 x) ; where x is the number
400
of tickets sold. Determine the maximum revenue possible.
4. A rectangular enclosure is formed with 300 feet of fencing with the barn wall as the fourth side. What is the
maximum area formed?
PRE-CALCULUS
QUADRATIC APPLICATIONS
NAME ____________________________
DATE ______________________
1. The sum of a number and its square is 56. Find the number
2. The sum of two numbers is 12. Find the two numbers if their product is to be a maximum.
3. Maximizing Revenue. The manufacturer of a gas clothes dryer has found that, when the unit price is p dollars,
the revenue R (in dollars) is: R( p)  4 p 2  400 p
What unit price should be established for the dryer to maximize revenue?
What is the maximum revenue?
4. Demand Equation. The price p and the quantity x sold of a certain product obey the demand equation
1
p   x  100;
6
0  x  600
a. Express the revenue as a function of x. (R = xp)
b. What is the revenue if 200 units are sold?
c. What quantity x maximizes revenue? What is the maximum revenue?
d. What price should the company charge to maximize revenue?
5. Enclosing a rectangular field. David has 400 yards available for fencing and wishes to enclose a rectangular
area.
a. Express the area A of the rectangle as a function of the width x of the rectangle.
b. For what value of x is the area largest?
c. What is the maximum area?
6. A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer
does not fence the side along the river, what is the largest area that can be enclosed? (See the figure)
7. Engineering. A suspension bridge has twin towers that extend 75 meters above the road surface and are 400
meters apart. The cables form a parabolic shape and are suspended from the tops of the towers. The cables touch
the road surface at the center of the bridge. Find the height of the cables 100 meters from the center.
8. A projectile is fired from a cliff 200 feet above the water and moves in the path of a parabola. The height of the
projectile is given by:
h( x ) 
 32 x 2
 x  200 , where x is the horizontal distance.
(50) 2
a. How far from the bridge is the projectile at a maximum height?
b. Find the maximum height.
c. How far from the base of the cliff will the projectile strike the water?