Algebra II Pre-AP
Warm-Up
Name:________________________
Twenty feet of fencing is provided to enclose a rectangular garden. The function that represents the
possible area of the garden is:
A  w  w 10  w
The graph of the Area as a function of the width of the rectangular garden is given below.
1.
Label the axes appropriately based on the situation
described.
2. In this situation, what is the reasonable interval for the
area of the garden? Is this the domain or range?
3. In this situation, what is the reasonable interval for the
possible widths of the garden? Is this the domain or range?
4. What is the area of the garden when the width is 2 feet?
5. What are the possible values for the width of the garden if
the area of the garden is 21 ft 2 ?
6. What is the maximum area that can be enclosed using the given fencing? What are the dimensions of the
rectangle that will enclose the garden with the maximum area?
Algebra II Pre AP
Worksheet – Quadratic Function Applications I
1. Corral Problem: A rectangular corral is to be built by stringing an electric fence as shown with y feet for the side
parallel to the river and x feet for each of the two sides perpendicular to the river. The total length of the fence
is to be 900 feet.
a. Write an equation expressing y in terms of x . What kind of function is this?
b. Let A(x) be the area of the corral in square feet. Write the particular equation of A(x).
What kind of function is this?
c. Find A(100) and A(300).
d. What value of x makes A(x) a maximum? What is this maximum area?
e. What values of x make A(x) equal 0?
f. Sketch the graph of function A(x). What is a reasonable domain?
2. Rectangular Field Problem: A rectangular field is 300 yards by 500 yards. A roadway of width x yards is to be
built inside the field. This problem concerns the region inside the roadway.
a. Write the length and width of the region as functions of x. What kind of functions are these?
b. Write the area of the region as a function of x. What kind of function is this?
c. Predict the area of the region if x = 5, 10, and 15.
d. What is the widest the roadway can be and still leave 100,000 square yards
in the region?
e. Sketch the graph of the area as a function of x in a reasonable domain.
f.
Find the value of x which makes the roadway have an area equal to the area of the inside region.
3. Car Insurance Problem: Suppose that you are an actuary for F. Bender's Insurance Agency. Your company
plans to offer a senior citizen's accident policy and you must predict the likelihood of an accident as a function of
the driver's age. From previous accident records, you find the following information:
Age (in years)
20
30
40
Accidents per 100 Million km Driven
440
280
200
You know that the number of accidents per 100 million km driven should reach a minimum then go up again for
very old drivers. Therefore, you assume that a quadratic function is a reasonable model.
a. Write the particular equation expressing accidents per 100 million kilometers in terms of age.
b. How many accidents per 100 million kilometers would you expect for an 80-year-old driver?
c. Based on your model, who is safer: a 16-year-old driver or a 70-year-old driver?
d. What age driver appears to be the safest?
e. You company decides to insure licensed drivers up to the age where the accident rate reaches 830 per million
km. What, then, is the domain of this quadratic function?
Algebra II Pre AP
EXTRA Practice: Quadratic Function Applications I
4. Artillery Problem: Artillerymen on a hillside are trying to hit a target behind a mountain on the other side of the
river (see figure below). Their cannon is at (x, y) = (3, 250), where x is in km and y is in m. The target is at (x, y) =
(-2, 50). In order to avoid hitting the mountain on the other side of the river, the projectile from the cannon
must go through the point (x, y) = (-1, 410).
a. Write the particular equation of the parabolic path of the projectile.
b. How high above the river will the projectile be where it crosses:
…the right riverbank, x = 2?
... the left riverbank, x = 0?
c. Approximately where will the projectile be when y = 130?
d. A reconnaissance plane is flying at 660 meters above the river Is it in danger of being hit by projectiles fired
along this parabolic path? Justify your answer.
5. Below is the number of feet required to stop and the speed at which the car is traveling prior to applying the
brakes.
Speed (mph)
Stopping distance (ft)
10
15
20
40
30
75
40
120
50
175
60
240
70
315
80
400
a) Write a quadratic regression equation for the data.
b) Predict the speed when the stopping distance is 500 feet.
c) You are traveling at 65 mph. You see a deer standing in the road about 250 feet in front of you. As soon as
you see the deer you hit your brakes. Will you stop in time or hit the deer? Explain.
For each of the following, define the variable(s), write a quadratic equation, and solve.
6. One number is equal to 6 times another number plus 7. The product of the two numbers is – 2. Find the
numbers.
7. Two numbers differ by 8 and the sum of their squares is 320. Find the two numbers.
8. Find two consecutive odd integers whose product is 195.
9. If a number is added to its square, the sum is 156. Find the number.
10. A rectangle is 15 cm wide and 18 cm long If both dimensions are decreased by the same amount, the area of
the new rectangle formed is 116 cm2 less than the area of the original. Find the dimensions of the new
rectangle.
11. The frame of a picture is 28 cm by 32 cm outside and is of uniform width. What is the width for the frame if 192
cm2 if the picture shows?
12. Vanessa built a rectangular pen for her dogs. She used an outside wall of the garage for one of the sides of the
pen. She had to buy 20 meters of fencing in order to build the other sides of the pen. Find the dimensions of
the pen if its area is 48 m2.
13. The total area of two square fields is 18,000 ft2. Each side of the larger field is 60 ft longer than a side of the
smaller field. Find the dimensions of the two fields.
14. A swimming pool is 20 feet wide and 30 feet long. It is surrounded by a concrete walkway that is x feet wide.
The area of the walkway is 216 ft2. What is the width of the walkway?
15. A rectangle is twice as long as it is wide. If its length is increased by 4 cm and its width is decreased by 3 cm, the
new rectangle formed has an area of 100 cm2. Find the dimensions of the original rectangle.