Download Examples for Problem Solving with Quadratic Equations

Sample Problems for “Problem Solving with Quadratic Equations” Lesson
(1)
A signal flare is fired into the air from a boat. The height of the flare in feet t seconds after it is
fired is h = -16t2 + 160t.
(a)
How high will the flare travel? When will it reach its maximum height?
400 feet, 5 seconds
When will the flare hit the water?
10 seconds
Explain how your answers to parts a and b could be found in a table and a graph of the
equation.
In a graph, the maximum point represents the maximum height of the flare, and
the right hand x-intercept represents the point at which the flare hits the water. In
a table, the entry when the height is its greatest represents the maximum height
reached by the flare, and the entry for when the height is once again 0 represents
the point at which the flare hits the water.
(b)
(c)
(2)
A square has sides of length x centimeters. A new rectangle is created by increasing one
dimension of the square by 3 centimeters.
(a)
How much greater is the area of the new rectangle than the area of the square?
3x
(b)
Write two equations for the area of the new rectangle.
A = x(x + 3)
A = x2 + 3x
(c)
Graph both equations on your calculator and copy the graphs onto your paper. Describe
the shapes of the graphs. How do they compare?
They form the same parabola.
(3)
KEY
The dot patterns below represent the first four square numbers 1, 4, 9, and 16.
1st
(a)
(b)
(c)
2nd
3rd
4th
What are the next two square numbers?
25, 36
Write an equation for calculating the nth square number.
s = n2
Make a table and a graph of the first ten square numbers. Describe the pattern of
change from one square number to the next.
Term Number
Number
of
Dots
1
1
2
4
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
The square numbers increase by consecutive odd integers beginning with 3 (3, 5, 7, …).
(4)
A cube with edges of length 2 centimeters is built from centimeter cubes. If you paint the faces
of this cube and then break it into centimeter cubes, how many cubes will be painted on three
faces? How many will be painted on two faces? On one face? How many will be unpainted?
(a)
Edge length
of large cube
2
3
4
5
6
(b)
(c)
Organize your data into a table like the one below.
Number of
cm cubes
8
27
64
125
216
Number of centimeter cubes painted on
3 faces
2 faces
1 face
0 faces
8
0
0
0
8
12
6
1
8
24
24
8
8
36
54
27
8
48
96
64
Study the patterns in the table. Look for a relationship between the edge length of the
large cube and the number of centimeter cubes. Tell whether the pattern change is
linear, quadratic, exponential, or none of these.
It would be none of these. The relationship involves cubing the length of the cube’s
edge.
Look for relationships between the edge length of the large cube and the number of
centimeter cubes painted on three faces, two faces, one face, and zero faces. Describe
each relationship you find and tell whether the pattern of change is linear, quadratic,
exponential, or none of these.
3 faces: Linear [y=8 regardless]
1 face: Quadratic [y = 6(x-2)2]
2 faces: Linear [y = 12(x-2)]
0 faces: None of these [ y = (x-2)3]