Chapter 1.5

Applications and Modeling with Quadratic Equations

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 1 Solving a Problem involving the Volume of a Box

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 2 Using the Pythogorean Theorem

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 3 Height of a Propelled Object

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 4 Analyzing Sport Utility Vehicle (SUV) Sales

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

Example 3 Height of a Propelled Object

0 = .016x2 + .124x – 2.213a

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Section 1.5 # 1 - 44