1 Ub CHAPTER 1 Equations and Inequalities SECTION 1.3 Complex Numbers; Quadratic Equations in the Complex Number System 1 09

Applications and Extensions99. College Tuition and Fees The average annual published

undergraduate tuition and-fee charges C, in dollars, for public four-year institutions from academic years 2000—2001through 2005 2006 can bç estimated by the equationC 20.2x2 + 314.5x + 3467.6, where x is the number ofyears afer the 2000—2001 academic year. Assuming the modelwill remain valid beyond 2005—2006, in what academic yearwill average annual tuition-and-fee charges be $8000?

Source: College Board, Trends in College Pricing 2005

100. Women’s Weekly Earnings The median weekly earnings E,in dollars, for full-time women wage and salary workers ages16 years and older from 2000 through 2004 can be estimatedby the equation E 0.14x2 + 7.8x.+ 540, where x is thenumber of years after 2000. Assuming the model will remainvalid beyond 2004, in what year will the median weekly earnings be $632?

Source. U.S. Department of Labor, Highlights of Women’sEarnings in 2004, September 2005

101. Dimensions of a Window The area of the opening of a rectangular window is to be 143 square feet. If the length is to be2 feet more than the width, what are the dimensions?

ular windowis to be 306 square centimeters. If the length exceeds thewidth by 1 centimeter, what are the dimensions?

103. Geometry Find the dimensions of a rectangle whoseperimeter is 26 meters and whose area is 40 square meters.

104. Watering a Field An adjustable water sprinkler that sprayswater in a circular pattern is placed at the center of a squarefield whose area is 1250 square feet (see the figure). What isthe shortest radius setting that can be used if the field is to becompletely enclosed within the circle?

0105. Constructing a Box An open box is to be constructed from

a square piece of sheet metal by removing a square of side1 foot from each corner and turning up the edges. If the boxis to hold 4 cubic feet, what should be the dimensions of thesheet metal?

106. Constructing a Box Rework Problem 105 if the piece ofsheet metal is a rectangle whose length is twice its width.

107. Physics A ball is thrown vertically upward from the top ofa building 96 feet tall with an initial velocity of 80 feet persecond. The distance s (in feet) of the ball from the groundafter t seconds is s 96 + 80t — 16t2.(a) After how many seconds does the ball strike the ground?(b) After how many seconds will the ball pass the top of

the building on its way down?

108. Physics An object is propelled vertically upward with aninitial velocity of 20 meters per second.The distances (in meters) of the object from the ground after t seconds iss 49~2 + 20t.

(b) When will it strike the ground?(c) Will the object reach a height of 100 meters?

109. Reducing the Size of a Candy Bar A jumbo chocolate barwith a rectangular shape measures 12 centimeters in length,7 centimeters in width, and 3 centimeters in thickness. Dueto escalating costs of cocoa, management decides to reducethe volume of the bar by 10°o.To accomplish this reduction,management decides that the new bar should have the same3 centimeter thickness, but the length and width of eachshould be reduced an equal number of centimeters. Whatshould be the dimensions of the new candy bar?

I

MILK cHOC0I~Afl

110. Reducing the Size of a Candy Bar Rework Problem 109 ifthe reduction is to be 20°c.

111. Constructing a Border around a Pool A circular pooi measures 10 feet across. One cubic yard of concrete is to be usedto create a circular border of uniform width around the pool.If the border is to have a depth of 3 inches, how wide will theborder be? (1 cubic yard 27 cubic feet) See the illustration.

112. Constructing a Border around a Pool Rework Problem 111if the depth of the border is 4 inches.

113. Constructing a Border around a Garden A landscaper, whojust completed a rectangular flower garden measuring 6 feetby 10 feet, orders 1 cubic yard of premixed cement, all ofwhich is to be used to create a border of uniform widtharound the garden. If the border is to have a depth of 3 inches,how wide will the border be? (1 cubic yard = 27 cubic feet)

J.~— lOft —~~jT6ft ~“~‘f ~

‘7.’ ..~

114. Dimensions of a Patio A contractor orders 8 cubic yards ofpremixed cement, all of which is to be used to pour a patiothat will be 4 inches thick. If the length of the patio is specified to be twice the width, what will be the patio dimensions?(1 cubic yard 27 cubic feet)

115. Comparing TVs The screen size of a television is determined by the length of the diagonal of the rectangular screen.

Traditional TVs come in a 4 : 3 format, meaning the ratio ofthe length to the width of the rectangular is 4 to 3. What is thearea of a 37-inch traditional TV screen? What is the area ofa 37-inch LCD TV whose screen is in a 16 : 9 format? Whichscreen is larger?

[Hint: If x is the length of a 4 : 3 format screen, then 4x isP the width.j

116. Comparing TVs Refer to Problem 115. Find the screen areaof a traditional 50-inch TV and compare it with a 50-inchPlasma TV whose screen is in a 16 : 9 format. Which screenis larger?

117. The sum of’the consecutive integers 1, 2, 3,.. . , n is given by

the formu~1~.n(n + 1). How many consecutive integers,

starting withil; must be added to get a sum of 666?

118. Geometry If a polygon of n sides has n n 3 diagonals,

how many ‘sid’es will a polygon with 65 diagonals have? Isthere a polygon with 80 diagonals?

121. Find k such that the equation kx2 + x + k 0 has arepeated real solution.

122. Find k such that the equation x kx 4 0 has arepeated real solution.

123. Show that the real solutions of the equationax2 + bx + c 0 are the negatives of the real solutions ofthe equation ax2 bx + c — 0. Assume that b2 — 4ac ≥ 0.

124. Show that the real solutions of the equationax2 + bx + c = 0 are the reciprocals of the real solutions ofthe equation cx2 + bx + a = 0. Assume that b2 — 4ac ≥ 0.

119. Show that the sum of the roots of a quadratic equation is

120. Show that the product of the roots of a quadratic equation

Isa

Discussion and Writing

~1

125. Which of ~ following pairs of equations are equivalent? 128. Create three quadratic equations: one having two distinct soExplain. lutions, one having no real solution, and one having exactly

(a) x2 9; 3x = 3 one real solution.(b) x = V’~; x = 3 129. The word quadratic seems to imply four (quad), yet a qua

(c) (x — 1)(x — 2) (x 1) ; x 2 x 1 dratic equation is an equation that involves a polynomial of

126. Descr.ilk three ways that you might solve a quadratic equa- degree 2. Investigate the origin of the term quadratic as it istion. State your preferred method; explain why you chose it. used in the expression quadratic equation. Write a brief essay

127. Explain tke ‘benefits of evaluating the discriminant of a qua- on your findings.dratic eq’uat~on before attempting to solve it.

‘Are You Prepared?’ Answers

3 3•

(x - 6~(x~ 1) 2. (2x 3)(x + 1) 3. { ~,3} 4. True

1.3 plex Numbers; Quadratic Equations° the Complex Number System*PREPARING FOR THIS SECTION Before getting started, review the following:

• Classification of Numbers (Section R.1, pp. 4—5) • Rationalizing Denominators (Section R.8, p. 74)

Now Work the ‘Are You Prepared?’ problems on page 116.

OBJECTIVES 1 Add, Subtract, Multiply, and Divide Complex Numbers (p. 110)

2 Solve Quadratic Equations in the Complex Number System (p. 114)

Complex NumbersOne property of a real number is that its square is nonnegative. For example, thereis no real number x for which

x 1

To remedy this situation, we introduce a new number called the imaginary unit.LCD16:9(a) When will the object be 15 meters above the ground?

traditional4:3 * This section may be omitted without any loss of continuity.