MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
141)
Hi–Tech UnWater begins a cable TV advertising campaign in Miami to market a new water. The percentage of
the target market that buys water is estimated by the function w(t) = 100(1 –e–0.02t), t represents the number
of days of the campaign. After how long will 90% of the target market have bought the water?
A)
120 days
B)
3 days
C)
90 days
D)
115 days
Answer:
142)
The number of books in a community college library increases according to the function B = 7200e0.03t, where t
is measured in years. How many books will the library have after 8 year(s)?
A)
9153
B)
10,275
C)
7200
D)
4462
Answer:
143)
Since life expectancy has increased in the last century, the number of Alzheimer‘s patients has increased
dramatically. The number of patients in the United States reached 4 million in 2000. Using data collected since
2000, it has been found that the data can be modeled by the exponential function y = 4.19549 ·(1.02531)x, where
x is the years since 2000. Estimate the Alzheimer’s patients in 2025. Round to the nearest tenth.
A)
3.9 million
B)
4.8 million
C)
8.0 million
D)
7.8 million
Answer:
144)
A sample of 800 grams of radioactive substance decays according to the function A(t) =800e–0.028t, where t is
the time in years. How much of the substance will be left in the sample after 10 years? Round to the nearest
whole gram.
A)
800 grams
B)
9 grams
C)
605 grams
D)
1 gram
Answer:
145)
The number of reports of a certain virus has increased exponentially since 1960. The current number of cases
can be approximated using the function r(t) = 207 e0.005t, where t is the number of years since 1960. Estimate
the of cases in the year 2010.
A)
266
B)
207
C)
240
D)
190
Answer:
146)
An initial investment of $12,000 is invested for 2 years in an account that earns 4% interest, compounded
quarterly. Find the amount of money in the account at the end of the period.
A)
$12,865.62
B)
$12,994.28
C)
$994.28
D)
$12,979.20
Answer:
147)
Suppose that $2200 is invested at 3% interest, compounded semiannually. Find the function for the amount of
money after t years.
A)
A = 2200 (1.015)2t
B)
A = 2200 (1.0125)2t
C)
A = 2200 (1.03)2t
D)
A = 2200 (1.015)t
Answer:
53
Use the REGRESSION feature on a graphing calculator.
148)
A strain of E–coli Beu–recA441 is placed into a petri dish at 30
Celsius and allowed to grow. The following data
are collected. Theory states that the number of bacteria in the petri dish will initially grow according to the law
of uninhibited growth. The population is measured using an optical device in which the amount of light that
passes through the petri dish is measured.
Time in hours , x Population, y
0 0.09
2.5 0.18
3.5 0.26
4.5 0.35
6 0.50
Find the exponential equation in the form y = a ·bx, where x is the hours of growth. Round to four decimal
places.
A)
y =1.3384x
B)
y = 1.3384 ·0.0903x
C)
y = 0.0903 ·1.3384x
D)
y =0.0903x
Answer:
149)
The total cost of the Democratic and the Republican national conventions has increased 596% over the 20–year
period between 1980 and 2004. The following table lists the total cost, in millions of dollars, for selected years.
Year, x Cost, y
1980, x = 0 $ 23.1
1984, x = 4 31.8
1988, x = 8 44.4
1992, x = 12 58.8
1996, x = 16 90.6
2000, x = 20 160.8
2004, x = 24 170.5
Find the exponential functions that best estimates this data. Round your answer to four decimal places
A)
y = 22.2887 · (1.0929)x
B)
y = 22.2887x· (1.0929)x
C)
y = 1.0929 · (22.2887)x
D)
y = 6.6643x + 2.8857
Answer:
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
150)
A particular bacterium is found to have a doubling time of 20 minutes. If a laboratory culture begins with a
population of 300 of this bacteria and there is no change in the growth rate, how many bacteria will be present
in 55 minutes? Use six decimal places in the interim calculation for the growth rate.
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Convert to a logarithmic equation.
151)
23= 8
A)
log2 3 = 8
B)
log8 2 = 3
C)
log2 8 = 3
D)
log3 8 = 2
Answer:
54
152)
52= 25
A)
2 =log 5 25
B)
5 =log2 25
C)
25 =log5 2
D)
2 =log25 5
Answer:
153)
100.4771 = 3
A)
3 = log 0.4771
B)
0.4771 = log 3
C)
0.4771 = log 10
D)
0.4771 =log 9 10
Answer:
154)
et= 7
A)
ln t = 7
B)
log 7 e = t
C)
ln 7 = t
D)
log 7 t = e
Answer:
Convert to an exponential equation.
155)
log9 27 =3
2
A)
27 =93/2
B)
27 =3
2
9
C)
3
2=927
D)
9 =273/2
Answer:
156)
log 8 512 = t
A)
t8= 512
B)
5128= t
C)
8512 = t
D)
8t= 512
Answer:
157)
ln 44 = 3.7842
A)
e44 = 3.7842
B)
e3.7842 = ln 44
C)
e3.7842 = 1
D)
e3.7842 = 44
Answer:
Evaluate.
158)
log884
A)
8
B)
32
C)
84
D)
4
Answer:
Use a calculator to evaluate the expression. Round the result to five decimal places.
159)
log 0.17
A)
–1.76955
B)
–0.76955
C)
–1.77196
D)
–4.07454
Answer:
160)
log 0.234
A)
–0.63074
B)
1.26364
C)
–1.45243
D)
0.234
Answer:
161)
log 51.237
A)
51.237
B)
3.93646
C)
1.70958
D)
Undefined
Answer:
55
162)
log (–10.25)
A)
2.32728
B)
1.01072
C)
–1.01072
D)
Undefined
Answer:
163)
log8 36.8
A)
1.73388
B)
1.56585
C)
0.57674
D)
3.60550
Answer:
164)
ln 0.027
A)
0.56864
B)
–3.61192
C)
–1.56864
D)
Undefined
Answer:
165)
ln 1097
A)
7.00033
B)
9.30292
C)
4.69775
D)
3.04021
Answer:
Write in terms of simpler forms.
166)
log4 XY
A)
log2 X +log2 Y
B)
log2 X –log2 Y
C)
log4 X +log4 Y
D)
log4 X –log4 Y
Answer:
167)
logb
b
y
A)
logbb– y
B)
logbb–logby
C)
logbb+logby
D)
log2b
b
y
Answer:
168)
logbM9
A)
9 logb M
B)
M +logb 9
C)
9 +logb M
D)
M logb 9
Answer:
169)
4a log4 b
A)
b4a
B)
a4b
C)
ba
D)
ab
Answer:
Solve for x to two decimal places (using a calculator).
170)
700 = 500(1.04)x
A)
8.58
B)
1.35
C)
520
D)
1.40
Answer:
171)
5.2 =1.00612x
A)
5.17
B)
22.97
C)
2.32
D)
1.07
Answer:
56
Use the properties of logarithms to solve.
172)
log7 x +log7(x – 2) =log7 24
A)
7
B)
24
C)
6
D)
2
Answer:
173)
logb x –logb 5 =logb 2 –logb(x – 3)
A)
3
B)
2, 5
C)
5
D)
2
Answer:
174)
logb(x + 3) +logb x =logb 54
A)
3
B)
6
C)
–6, –3
D)
–6
Answer:
175)
log 6 (4x – 5) = 1
A)
log 5
4
B)
7
C)
11
6
D)
11
4
Answer:
176)
ln (3x – 4) = ln 20 – ln (x – 5)
A)
–5, –19
3
B)
5, 5
3
C)
0, 19
3
D)
19
3
Answer:
177)
log (x + 10) – log (x + 4) = log x
A)
–5
B)
6
C)
2, – 5
D)
2
Answer:
178)
log (x – 9) = 1 – log x
A)
10
B)
–10
C)
–1, 10
D)
–10, 1
Answer:
Graph by converting to exponential form first.
179)
y =log 4(x – 2)
57
A)
B)
C)
D)
Answer:
180)
y =log 2(x + 1)
58
A)
B)
C)
D)
Answer:
Graph the function using a calculator and point–by–point plotting. Indicate increasing and decreasing intervals.
181)
f(x) = 3 ln x
59
A)
Increasing: (0,
)
B)
Decreasing: (0,
)
C)
Decreasing: (0,
)
D)
Increasing: (–3,
)
Answer:
182)
f(x) = – 3 ln x
60
A)
Decreasing: (0, –3]
Increasing: [–3,
)
B)
Decreasing: 0, 1
2
Increasing: 1
2,
C)
Decreasing: (0,
)
D)
Decreasing: (0, 1]
Increasing: [1,
)
Answer:
183)
f(x) = – 4– ln x
61
A)
Decreasing: (0,
)
B)
Decreasing: (0,
)
C)
Increasing (–4,
)
D)
Increasing (0,
)
Answer:
184)
f(x) = 2 – ln(x + 4)
62
A)
Decreasing: (0,
)
B)
Decreasing: (4,
)
C)
Decreasing: (–4,
)
D)
Decreasing: (–4,
)
Answer:
Solve the problem.
185)
If $1250 is invested at a rate of 8 1
4% compounded monthly, what is the balance after 10 years?
[A =P(1 + i)n]
A)
$1594.31
B)
$1031.25
C)
$2281.25
D)
$2844.31
Answer:
186)
If $4,000 is invested at 7% compounded annually, how long will it take for it to grow to $6,000, assuming no with
are made? Compute answer to the next higher year if not exact.
[A =P(1 + r)t]
A)
2 years
B)
5 years
C)
8 years
D)
6 years
Answer:
187)
In North America, coyotes are one of the few species with an expanding range. The future population of coyotes
in a region of Mississippi valley can be modeled by the equation P = 59 + 12 · ln(18t + 1), where t is time in
years. Use the equation to determine when the population will reach 170. (Round your answer to the nearest
tenth year.)
A)
578.0 years
B)
586.2 years
C)
583.1 years
D)
581.3 years
Answer:
63
188)
A country has a population growth rate of 2.4% compounded continuously. At this rate, how long will it take
for the population of the country to double? Round your answer to the nearest tenth.
A)
.29 years
B)
30 years
C)
2.9 years
D)
28.9 years
Answer:
189)
A carbon–14 dating test is performed on a fossil bone, and analysis finds that 15.5% of the original amount of
carbon–14 is still present in the bone. Estimate the age of the fossil bone. (Recall that carbon–14 decays
according to the equation A =A0e–0.000124t).
A)
1,500 years
B)
15, 000 years
C)
15,035 years
D)
150 years
Answer:
190)
Assume that a savings account earns interest at the rate of 2% compounded monthly. If this account contains
$1000 now, how many months will it take for this amount to double if no withdrawals are made?
A)
417 months
B)
12 months
C)
408 months
D)
450 months
Answer:
191)
U. S. Census Bureau data shows that the number of families in the United States (in millions) in year x is given
by h(x) = 51.42 + 15.473 · log x , where x = 0 is 1980. How many families were there in 2002?
A)
48 million
B)
72 million
C)
21 million
D)
90 million
Answer:
192)
The level of a sound in decibels (db) is determined by the formula N = 10 · log(I ×1012) db, where I is the
intensity of the sound in watts per square meter. A certain noise has an intensity of 8.49 ×10–4 watts per square
meter. What is the sound level of this noise? (Round your answer to the nearest decibel.)
A)
89 db
B)
206 db
C)
79 db
D)
9 db
Answer:
193)
Book sales on the Internet (in billions of dollars) in year x are approximated by f(x) = 1.84 + 2.1 · ln x, where x =
0 corresponds to 2000. How much will be spent on Internet book sales in 2008? Round to the nearest tenth.
A)
8.0 billion
B)
6.2 billion
C)
6.0 billion
D)
3.9 billion
Answer:
64