Advanced Algebra and Trig 4th Nine Weeks Exam Review
There are 51 questions on this review. Your test on Wednesday, May 4, will be 25 MC questions with
the exception of 1 scatter plot at the end. You will only have 3 tests total this nine weeks which
make up 50% of your grade, therefore, study. You WILL turn in this test review, with all work shown
on notebook paper.
Solve the problem.
1) The rabbit population in a forest area grows at the rate of 7% monthly. If there are 220 rabbits in April, find
how many rabbits (rounded to the nearest whole number) should be expected by next April. Use
y = 220(2.7) 0.07t.
2) A city is growing at the rate of 0.8% annually. If there were 4,566,000 residents in the city in 1993, find
how many (to the nearest ten-thousand) are living in that city in 2000. Use y = 4,566,000(2.7) 0.008t.
nt
Use the compound interest formulas A = P 1 + r
and A = Pert to solve.
n
3) Find the accumulated value of an investment of $16,000 at 4% compounded semiannually for 7 years.
4) Find the accumulated value of an investment of $200 at 8% compounded quarterly for 4 years.
5) Find the accumulated value of an investment of $5000 at 5% compounded monthly for 8 years.
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible,
evaluate logarithmic expressions without using a calculator.
5
6) ln ey
81
x-1
7) log 3
8
8) log b xy
z7
9) ln
x
y
9
10) log
19
14
q 2p
5
11) log
9
4
m n
k2
3
4 x+5
12) log a x
(x -2)2
1
Use properties of logarithms to condense the logarithmic expression. Write the expression as a single
logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions.
13) 7ln (x - 4) - 8 ln x
14) ( log
15) 3 log
a
6
q - log
a
x + 5 log
r) + 3 log
6
a
p
(x - 6)
16) 1 (log 3 (r - 6) - log 3 r)
2
17) 2 log 4 3 + 1 log 4 (r - 3) - 1 log 4 r
5
2
18) 1 (log 4 x + log 4 y)
4
19) 1 (log 6 x + log 6 y) - 3 log 6 (x + 6)
5
20) 1 [4ln (x + 8) - ln x - ln (x2 - 5)]
5
Solve the equation.
21) 2(3x - 5) = 16
22) 32x = 8
23) 10(x - 8)/8 =
24) 1024x = 1
10
4
25) 4x + 5 = 8x - 1
26) ex + 5 = 1
e9
Solve the exponential equation. Express the solution set in terms of natural logarithms.
27) e2x = 6
28) e x + 6 = 8
29) 4x + 4 = 52x + 5
2
Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal
places, for the solution.
30) e2x = 7
31) e x + 2 = 4
32) 7x = 6x + 7
33) e2x - 8 - 6 = 1219
34) e2x + ex - 6 = 0
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original
logarithmic expressions. Give the exact answer.
35) log (x - 2) = -3
4
36) log
4
(x + 2) + log
4
(x - 4) = 2
37) 6 ln (3x) = 30
38) 5 + 7 ln x = 13
39) ln 3 + ln (x - 1) = 0
40) log
Solve.
5
(6x + 5) = log
5
(6x + 8)
41) The function A = A 0e-0.00866x models the amount in pounds of a particular radioactive material stored in a
concrete vault, where x is the number of years since the material was put into the vault. If 200 pounds of
the material are initially put into the vault, how many pounds will be left after 90 years?
42) The function A = A 0e-0.01155x models the amount in pounds of a particular radioactive material stored in a
concrete vault, where x is the number of years since the material was put into the vault. If 600 pounds of
the material are placed in the vault, how much time will need to pass for only 106 pounds to remain?
43) The value of a particular investment follows a pattern of exponential growth. In the year 2000, you invested
money in a money market account. The value of your investment t years after 2000 is given by the
exponential growth model A = 8000e0.052t. How much did you initially invest in the account?
44) The value of a particular investment follows a pattern of exponential growth. In the year 2000, you invested
money in a money market account. The value of your investment t years after 2000 is given by the
exponential growth model A = 4200e0.058t. When will the account be worth $5297?
45) The half-life of silicon-32 is 710 years. If 40 grams is present now, how much will be present in 500 years?
(Round your answer to three decimal places.)
46) A fossilized leaf contains 32% of its normal amount of carbon 14. How old is the fossil (to the nearest year)?
Use 5600 years as the half-life of carbon 14.
3
Solve the problem.
47) The logistic growth function f(t) =
440
describes the population of a species of butterflies
1 + 10.0e-0.29t
t months after they are introduced to a non-threatening habitat. How many butterflies were initially
introduced to the habitat?
48) The logistic growth function f(t) =
800
describes the population of a species of butterflies
1 + 7.0e-0.21t
t months after they are introduced to a non-threatening habitat. What is the limiting size of the butterfly
population that the habitat will sustain?
49) The logistic growth function f(t) =
680
describes the population of a species of butterflies
1 + 5.8e-0.12t
t months after they are introduced to a non-threatening habitat. How many butterflies are expected in the
habitat after 20 months?
Present data in the form of tables. For the data set shown by the table,
a. Create a scatter plot for the data.
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best
choice for modeling the data.
50) Number of Homes Built in a Town by Year
Year Number of Homes
1985
12
1991
92
1994
145
1997
192
2002
224
y
x
4
51) Percentage of Population Living in the
South Suburbs of a Large City
Year Percent
1950 55
1960 70
1970 73
1980 76
2000 76
y
x
5
Answer Key
Testname: AAT 4TH NINE WEEKS TEST REVIEW
507
4,830,000
$21,111.66
$274.56
$7452.93
6) 1 ln y + 1
5
5
1)
2)
3)
4)
5)
7) 4 - 1 log 3(x - 1)
2
8) log b x + 8log b y - 7log b z
9) 1 ln x - 1 ln y
2
2
10) 1 log 19 14 - 2 log 19 q 9
log
p
19
11) 1 log 9 m + 1 log 9 n - 2
5
4
log
k
9
12) 4 log a x + 1 log a (x + 5) - 2
3
log a (x - 2)
7
13) ln (x - 4)
8
x
14) log
15) log
qp 3
r
a
x3(x - 6)5
6
r-6
r
16) log 3
17) log 4
18) log 4
9
4
5
51) a.
y
140
28) {ln 8 - 6}
29) 5 ln 5 - 4 ln 4
ln 4 - 2 ln 5
120
100
80
0.97
-0.61
81.36
7.56
0.69
35) 129
64
30)
31)
32)
33)
34)
60
40
20
1940 1950 1960 1970 1980 1990
36) {6}
5
37) e
3
b. Logarithmic Function
e 8/7
39) { 4 }
3
38)
40)
41)
42)
43)
44)
45)
46)
47)
48)
49)
50)
∅
92 pounds
150 years
$8000.00
2004
24.551
9189
40 butterflies
800 butterflies
446 butterflies
a.
220
y
200
r-3
r
180
160
140
xy
120
5
100
xy
19) log 6
(x + 6)3
4
20) ln 5 (x + 8)
2
x(x - 5)
21) {3}
22) 3
5
26) {-14}
27) ln 6
2
80
60
40
20
1980
1985
1990
b. Exponential function
23) {12}
24) - 1
10
25) {13}
6
1995
2000
x
x