College Algebra 9th txtbk

530 CHAPTER 8 SEQUENCES, INDUCTION, AND PROBABILITY
APPLICATIONS
75. BUSINESS In investigating different job opportunities, you find
that firm A will start you at $25,000 per year and guarantee you a
raise of $1,200 each year whereas firm B will start you at $28,000
per year but will guarantee you a raise of only $800 each year. Over
a period of 15 years, how much would you receive from each firm?
76. BUSINESS In Problem 75, what would be your annual salary at
each firm for the tenth year?
77. ECONOMICS The government, through a subsidy program, distributes
$1,000,000. If we assume that each individual or agency
spends 0.8 of what is received, and 0.8 of this is spent, and so on, how
much total increase in spending results from this government action?
78. ECONOMICS Because of reduced taxes, an individual has an
extra $600 in spendable income. If we assume that the individual
spends 70% of this on consumer goods, that the producers of these
goods in turn spend 70% of what they receive on consumer goods,
and that this process continues indefinitely, what is the total amount
spent on consumer goods?
79. BUSINESS If $P is invested at r% compounded annually, the
amount A present after n years forms a geometric sequence with a
common ratio 1 r. Write a formula for the amount present after n
years. How long will it take a sum of money P to double if invested
at 6% interest compounded annually?
80. POPULATION GROWTH If a population of A 0 people grows at
the constant rate of r% per year, the population after t years forms
a geometric sequence with a common ratio 1 r. Write a formula
for the total population after t years. If the world’s population is increasing
at the rate of 2% per year, how long will it take to double?
81. FINANCE Eleven years ago an investment earned $7,000 for the
year. Last year the investment earned $14,000. If the earnings from
the investment have increased the same amount each year, what is
the yearly increase and how much income has accrued from the investment
over the past 11 years?
82. AIR TEMPERATURE As dry air moves upward, it expands. In so
doing, it cools at the rate of about 5°F for each 1,000-foot rise. This
is known as the adiabatic process.
(A) Temperatures at altitudes that are multiples of 1,000 feet form
what kind of a sequence?
(B) If the ground temperature is 80°F, write a formula for the temperature
T n in terms of n, if n is in thousands of feet.
83. ENGINEERING A rotating flywheel coming to rest rotates 300
revolutions the first minute (see the figure). If in each subsequent
minute it rotates two-thirds as many times as in the preceding minute,
how many revolutions will the wheel make before coming to rest?
84. PHYSICS The first swing of a bob on a pendulum is 10 inches.
If on each subsequent swing it travels 0.9 as far as on the preceding
swing, how far will the bob travel before coming to rest?
85. FOOD CHAIN A plant is eaten by an insect, an insect by a
trout, a trout by a salmon, a salmon by a bear, and the bear is eaten
by you. If only 20% of the energy is transformed from one stage
to the next, how many calories must be supplied by plant food to
provide you with 2,000 calories from the bear meat?
86. GENEALOGY If there are 30 years in a generation, how many direct
ancestors did each of us have 600 years ago? By direct ancestors
we mean parents, grandparents, great-grandparents, and so on.
87. PHYSICS An object falling from rest in a vacuum near the surface
of the Earth falls 16 feet during the first second, 48 feet during
the second second, 80 feet during the third second, and so on.
(A) How far will the object fall during the eleventh second?
(B) How far will the object fall in 11 seconds?
(C) How far will the object fall in t seconds?
88. PHYSICS In Problem 87, how far will the object fall during:
(A) The twentieth second? (B) The tth second?
89. BACTERIA GROWTH A single cholera bacterium divides every
1
2 hour to produce two complete cholera bacteria. If we start with a
colony of A 0 bacteria, how many bacteria will we have in t hours,
assuming adequate food supply?
90. CELL DIVISION One leukemic cell injected into a healthy
1
mouse will divide into two cells in about 2 day. At the end of the day
these two cells will divide again, with the doubling process continuing
each 2 day until there are 1 billion cells, at which time the mouse
1
dies. On which day after the experiment is started does this happen?
91. ASTRONOMY Ever since the time of the Greek astronomer
Hipparchus, second century B.C., the brightness of stars has been
measured in terms of magnitude. The brightest stars, excluding the
sun, are classed as magnitude 1, and the dimmest visible to the eye
are classed as magnitude 6. In 1856, the English astronomer N. R.
Pogson showed that first-magnitude stars are 100 times brighter
than sixth-magnitude stars. If the ratio of brightness between consecutive
magnitudes is constant, find this ratio. [Hint: If b n is the
brightness of an nth-magnitude star, find r for the geometric sequence
b 1 , b 2 , b 3 , . . . , given b 1 100b 6 .]
92. PUZZLE If a sheet of very thin paper 0.001-inch thick is torn in
half, and each half is again torn in half, and this process is repeated
for a total of 32 times, how high will the stack of paper be if the
pieces are placed one on top of the other? Give the answer to the
nearest mile.