Chapter 1.5
Applications and Modeling with
Quadratic Equations
Example 1 Solving a Problem involving the Volume of a Box
A piece of machinery is capable of producing
rectangular sheets of metal such that the length is
three times the width. Furthermore, equal-sized
squares measuring 5 in. on a side can be cut from
the corners so that the resulting piece of metal can
be shaped into an open ox by folding up the flaps.
If specifications call for the volume of the box to
be 1435 in2, what should the dimensions of the
original piece of metal be?
Example 1 Solving a Problem involving the Volume of a Box
A piece of machinery is capable of producing
rectangular sheets of metal such that the length is
three times the width.
Example 1 Solving a Problem involving the Volume of a Box
A piece of machinery is capable of producing
rectangular sheets of metal such that the length is
three times the width. Furthermore, equal-sized
squares measuring 5 in. on a side can be cut from
the corners so that the resulting piece of metal can
be shaped into an open ox by folding up the flaps.
Example 1 Solving a Problem involving the Volume of a Box
A piece of machinery is capable of producing
rectangular sheets of metal such that the length is
three times the width. Furthermore, equal-sized
squares measuring 5 in. on a side can be cut from
the corners so that the resulting piece of metal can
be shaped into an open ox by folding up the flaps.
If specifications call for the volume of the box to
be 1435 in2, what should the dimensions of the
original piece of metal be?
Example 1 Solving a Problem involving the Volume of a Box
A piece of machinery is capable of producing
rectangular sheets of metal such that the length is
three times the width. Furthermore, equal-sized
squares measuring 5 in. on a side can be cut from
the corners so that the resulting piece of metal can
be shaped into an open ox by folding up the flaps.
If specifications call for the volume of the box to
be 1435 in2, what should the dimensions of the
original piece of metal be?
Example 1 Solving a Problem involving the Volume of a Box
Volume = length x width x height
1435 = (3x-10) (x – 10) (5)
Example 2 Using the Pythogorean Theorem
Erik Van Erden finds a piece of property in the
shape of a right triangle. To get some idea of its
dimensions, he measures the three sides, starting
with the shortest side. He finds that the longer leg
is 20 m longer than twice the length of the shorter
leg. They hypotunuse is 10 m longer than the
length of the longer leg. Find the lengths of the
sides of the triangular lot.
Example 2 Using the Pythogorean Theorem
Erik Van Erden finds a piece of property in the
shape of a right triangle. To get some idea of its
dimensions, he measures the three sides, starting
with the shortest side. He finds that the longer leg
is 20 m longer than twice the length of the shorter
leg. They hypotunuse is 10 m longer than the
length of the longer leg. Find the lengths of the
sides of the triangular lot.
Example 3 Height of a Propelled Object
If a projectile is shot vertically upward from the
ground with an initial velocity of 100 ft per
second, neglecting air resistance, its height s (in
feet) above the ground t seconds after projection is
given by
s = -16t2 + 100t
(a)After how many seconds will it be 50 ft above
the ground?
Example 3 Height of a Propelled Object
If a projectile is shot vertically upward from the
ground with an initial velocity of 100 ft per
second, neglecting air resistance, its height s (in
feet) above the ground t seconds after projection is
given by
s = -16t2 + 100t
(b)How long will it take for the projectile to return
to the ground?
Example 4 Analyzing Sport Utility Vehicle (SUV) Sales
The bar graph in Figure 8 shows sales of SUVs
(sport utility vehicles) in the United States, in
millions. The quadratic equation
S = .016x2 + .124x + .787
models sales of SUVs from 1990 to 2001, where S
represents sales in millions, and x = 0 represents
1990, x = 1 represents 1991, and so on.
Example 3 Height of a Propelled Object
S = .016x2 + .124x + .787
(a) Use the model to determine sales in 2000 and
2001. Compare the results to the actual
figures of 3.4 million and 3.8 million from the
graph.
S = .016(10)2 + .124(10) + .787
S = .016(100) + .124(10) + .787
S=
1.6 + 1.24+ .787
S=
3.627
Example 3 Height of a Propelled Object
S = .016x2 + .124x + .787
(a) Use the model to determine sales in 2000 and
2001. Compare the results to the actual
figures of 3.4 million and 3.8 million from the
graph.
S = .016(11)2 + .124(11) + .787
S = .016(121) + .124(11) + .787
S=
1.936 + 1.364+ .787
S=
4.087
Example 3 Height of a Propelled Object
S = .016x2 + .124x + .787
(b) According to the model, in what year do sales
reach 3 million? (Round down to the nearest
year.) Is the result accurate?
3 = .016x2 + .124x + .787
0 = .016x2 + .124x – 2.213
0 = .016x2 + .124x – 2.213
 b  b  4ac
x
2a
2
 (.124 )  ( .124 ) 2  4( .016 )( - 2.213 )
x
2( .016 )
x
 .124 
.015376 ( .064)( - 2.213)
.032
 .124  .157  .124  .3962323

x
.032
.032
0 = .016x2 + .124x – 2.213
 .124  .157
x
.032
 .124  .3962323

.032
 .124  .3962323
or
.032
 .124  .3962323

.032
x  16257258
or
x  8.5072594
Section 1.5 # 1 - 44