6.5E: Exercises
 Page ID
 30869
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Additional Factoring Applications
Number problems
61. The product of two consecutive even integers is \(624\). Find the integers.
 Answer

\(26\) and \(−24\), and \(24\) and \(26\)
62. The product of two consecutive even integers is \(528\). Find the integers.
63. The product of two consecutive positive integers is \(552\). Find the integers.
 Answer

\(23\), \(24\)
64. The product of two consecutive positive integers is \(756\). Find the integers..
65.
The product of two consecutive odd integers is \(483\). Find the integers.
 Answer

\(−23\) and \(−21\), and \(21\) and \(23\)
66. The product of two consecutive odd integers is \(783\). Find the integers.
67. You have two positive numbers. The second number is three more than two times the ﬁrst number. The diﬀerence of their squares is \(144\). Find both positive numbers.
 Answer

\(5\) and \(13\)
68. You have two positive numbers. The second number is two more than three times the ﬁrst number. The diﬀerence of their squares is \(60\). Find both positive numbers.
69.
Two numbers diﬀer by \(5\). The sum of their squares is \(97\). Find the two numbers.
 Answer

\(4\) and \(9\), and \(−4\) and \(−9\)
70. Two numbers diﬀer by \(6\). The sum of their squares is \(146\). Find the two numbers.
Area of Rectangles
71. A rectangular canvas picture measures \(14\) inches by \(36\) inches. The canvas is mounted inside a frame of uniform width, increasing the total area covered by both canvas and frame to \(720\) square inches. Find the uniform width of the frame.
 Answer

\(2\) inches
72. A rectangular canvas picture measures \(10\) inches by \(32\) inches. The canvas is mounted inside a frame of uniform width, increasing the total area covered by both canvas and frame to \(504\) square inches. Find the uniform width of the frame.
73. A rectangle has perimeter \(42\) feet and area \(104\) square feet. Find the dimensions of the rectangle.
 Answer

\(8\) feet by \(13\) feet
74. A rectangle has perimeter \(32\) feet and area \(55\) square feet. Find the dimensions of the rectangle.
75. The length of a rectangle is three feet longer than six times its width. If the area of the rectangle is \(165\) square feet, what is the width of the rectangle?
 Answer

\(5\) feet
76. The length of a rectangle is three feet longer than nine times its width. If the area of the rectangle is \(90\) square feet, what is the width of the rectangle?
77. The ratio of the width to the length of a given rectangle is \(2\) to \(3\), or \(\dfrac {2}{3}\). If the width and length are both increased by \(4\) inches, the area of the resulting rectangle is \(80\) square inches. Find the width and length of the original rectangle.
 Answer

\(4\) inches by \(6\) inches
78. The ratio of the width to the length of a given rectangle is \(3\) to \(4\), or \(\dfrac {3}{4}\). If the width is increased by \(3\) inches and the length is increased by \(6\) inches, the area of the resulting rectangle is \(126\) square inches. Find the width and length of the original rectangle.
Area of Triangles
79. Find the area of \(\triangle ABC\) 
80. Find the area of \(\triangle ABC\) x 
 Answer

79. \(x=24,\) area is \(240\) square units
81. Find the value of \(x\) given the area of \(\triangle ABC\) is 35:  82. Find the value of \(x\) given the area of \(\triangle ABC\) is 24. 
 Answer

81. \(x=5\)
83. Find the value of \(x\) given the area of \(\triangle ABC\) is 12:  84. Find the value of \(x\) given the area of \(\triangle ABC\) is 108: 
 Answer

83. \(x=4\)
Area of circles
85. The radius of the outer circle is one inch longer than twice the radius of the inner circle.
If the area of the shaded region is \(40\pi \) square inches, what is the length of the inner radius?
 Answer

\(3\) inches
86. The radius of the outer circle is two inches longer than three times the radius of the inner circle.
If the area of the shaded region is \(180\pi \) square inches, what is the length of the inner radius?
87. Find the area of the circle to the nearest tenth 
88. Find the area of the circle to the nearest tenth 
 Answer

87. \(78.5\) square units
Projectile Problems
89. A projectile is ﬁred at an angle into the air from atop a cliﬀ overlooking the ocean. The projectile’s distance (in feet) from the base of the cliﬀ is given by the equation \(x = 180t \) and the projectile’s height above sea level (in feet) is given by the equation \(y = −16t^2 + 352t + 1664 \) where \(t\) is the amount of time (in seconds) that has passed since the projectile’s release. How much time passes before the projectile splashes into the ocean? At that time, how far is the projectile from the base of the cliﬀ?
 Answer

\(26\) seconds, \(4,680\) feet
90. A projectile is ﬁred at an angle into the air from atop a cliﬀ overlooking the ocean. The projectile’s distance (in feet) from the base of the cliﬀ is given by the equation \(x = 140t \) and the projectile’s height above sea level (in feet) is given by the equation \(y = −16t^2 + 288t + 1408 \) where \(t\) is the amount of time (in seconds) that has passed since the projectile’s release. How much time passes before the projectile splashes into the ocean? At that time, how far is the projectile from the base of the cliﬀ?