College Algebra 9th txtbk

146 CHAPTER 2 GRAPHS
Problems 67–72 refer to the quadrilateral with vertices A (0, 2),
B (4, 1), C (1, 5), and D (3, 2).
67. Show that AB DC. 68. Show that DA CB.
69. Show that AB BC. 70. Show that AD DC.
71. Find an equation of the perpendicular bisector* of AD.
72. Find an equation of the perpendicular bisector of AB.
73. Prove that if a line L has x intercept (a, 0) and y intercept (0, b),
then the equation of L can be written in the intercept form
74. Prove that if a line L passes through P 1 (x 1 , y 1 ) and
P 2 (x 2 , y 2 ), then the equation of L can be written in the twopoint
form
( y y 1 )(x 2 x 1 ) ( y 2 y 1 )(x x 1 )
Problems 75–80 are calculus related. Recall
that a line tangent to a circle at a point is
perpendicular to the radius drawn to that
point (see the figure). Find the equation of the
line tangent to the circle at the indicated point.
Write the final answer in the standard form
Ax By C, A 0. Graph the circle and the
tangent line on the same coordinate system.
75. x 2 y 2 25, (3, 4) 76. x 2 y 2 100, (8, 6)
77. x 2 y 2 50, (5, 5) 78. x 2 y 2 80, (4, 8)
79. (x 3) 2 ( y 4) 2 169, (8, 16)
80. (x 5) 2 ( y 9) 2 289, (13, 6)
APPLICATIONS
x
a y b 1 a, b 0
81. BOILING POINT OF WATER At sea level, water boils when it
reaches a temperature of 212°F. At higher altitudes, the atmospheric
pressure is lower and so is the temperature at which water
boils. The boiling point B in degrees Fahrenheit at an altitude of x
feet is given approximately by
B 212 0.0018x
(A) Complete Table 3.
Table 3
x 0 5,000 10,000 15,000 20,000 25,000 30,000
B
(B) Based on the information in the table, write a brief verbal description
of the relationship between altitude and the boiling point of water.
82. AIR TEMPERATURE As dry air moves upward, it expands and
cools. The air temperature A in degrees Celsius at an altitude of x
kilometers is given approximately by
A 25 9x
*The perpendicular bisector of a line segment is a line perpendicular to
the segment and passing through its midpoint.
(A) Complete Table 4.
Table 4
x 0 1 2 3 4 5
A
(B) Based on the information in the table, write a brief verbal
description of the relationship between altitude and air temperature.
83. COST ANALYSIS A doughnut shop has a fixed cost of $124 per
day and a variable cost of $0.12 per doughnut. Find the total daily
cost of producing x doughnuts. How many doughnuts can be produced
for a total daily cost of $250?
84. COST ANALYSIS A small company manufactures picnic tables.
The weekly fixed cost is $1,200 and the variable cost is $45 per table.
Find the total weekly cost of producing x picnic tables. How many
picnic tables can be produced for a total weekly cost of $4,800?
85. PHYSICS Hooke’s law states that the relationship between the
stretch s of a spring and the weight w causing the stretch is linear
(a principle upon which all spring scales are constructed). For a particular
spring, a 5-pound weight causes a stretch of 2 inches, while
with no weight the stretch of the spring is 0.
(A) Find a linear equation that expresses s in terms of w.
(B) What is the stretch for a weight of 20 pounds?
(C) What weight will cause a stretch of 3.6 inches?
86. PHYSICS The distance d between a fixed spring and the floor is
a linear function of the weight w attached to the bottom of the
spring. The bottom of the spring is 18 inches from the floor when
the weight is 3 pounds and 10 inches from the floor when the
weight is 5 pounds.
(A) Find a linear equation that expresses d in terms of w.
(B) Find the distance from the bottom of the spring to the floor if no
weight is attached.
(C) Find the smallest weight that will make the bottom of the spring
touch the floor. (Ignore the height of the weight.)
87. PHYSICS The two most widespread temperature scales are
Fahrenheit* (F) and Celsius † (C). It is known that water freezes at
32°F or 0°C and boils at 212°F or 100°C.
(A) Find a linear equation that expresses F in terms of C.
(B) If a European family sets its house thermostat at 20°C, what is
the setting in degrees Fahrenheit? If the outside temperature in Milwaukee
is 86°F, what is the temperature in degrees Celsius?
88. PHYSICS Two other temperature scales, used primarily by scientists,
are Kelvin ‡ (K) and Rankine** (R). Water freezes at 273 K
or 492°R and boils at 373 K or 672°R. Find a linear equation that
expresses R in terms of K.
89. OCEANOGRAPHY After about 9 hours of a steady wind, the
height of waves in the ocean is approximately linearly related to
*Invented in 1724 by Daniel Gabriel Fahrenheit (1686–1736), a German
physicist.
† Invented in 1742 by Anders Celsius (1701–1744), a Swedish astronomer.
‡ Invented in 1848 by Lord William Thompson Kelvin (1824–1907), a
Scottish mathematician and physicist. Note that the degree symbol “ ° ”
is not used with degrees Kelvin.
**Invented in 1859 by John Maquorn Rankine (1820–1872), a Scottish
engineer and physicist.