Solve the problem.
48)
A bacterial culture grows exponentially; that is, P(t) = 100ekt, where P(t) is the size of the culture at
time t hours. Suppose that after 2 hours the size of the culture is 400. What is k (approximately)?
48)
A)
0.69
B)
0.06
C)
0.48
D)
3
E)
none of these
49)
A beverage company works out a demand function for its sale of soda and finds it to be
q = D(x) =3500 –24x,
where q = the quantity of sodas sold when the price per can, in cents, is x. At a price of 110 cents
per can, will a small increase in price cause the total revenue to increase, decrease, or stay the same?
49)
A)
Decrease
B)
Stay the same
C)
Increase
A
50)
In a town of 10,000 people, the number of people who during each day first hear the news of a local
tax increase is one–tenth the number of people who have not yet heard the news. If f(t) stands for
the number of informed people in the town, what is the differential equation f(t) satisfies?
50)
A)
f(t) =10f'(t)
10,000
B)
f'(t) =1
10 f(t)
C)
f'(t) =1
10 (10,000 – f(t))
D)
f'(t) +1
10 f(t) = 10,000
E)
none of these
C
A
51)
Determine the percentage rate of change of f(x) =e0.9x at x = 15 and x = 30.
51)
A)
f'(15)
f(15) = 17%, f'(30)
f(30) = 29%
B)
f'(15)
f(15) = 90%, f'(30)
f(30) = 90%
C)
f'(15)
f(15) = 22%, f'(30)
f(30) = 5%
D)
f'(15)
f(15) = 38%, f'(30)
f(30) = 14%
52)
How much money has to be invested now at 8% continuous interest in order to have $1000 after 5
years?
52)
A)
$670.32
B)
$18.32
C)
$183.20
D)
$461.56
E)
none of these
53)
Determine the percentage rate of change of h(p) =4
7p + 3 at p = 1 and p = 9.
53)
A)
h'(1)
h(1) = – 40%, h'(9)
h(9) = – 6%
B)
h'(1)
h(1) = 40%, h'(9)
h(9) = 6%
C)
h'(1)
h(1) = – 70%, h'(9)
h(9) = – 11%
D)
h'(1)
h(1) = 70%, h'(9)
h(9) = 11%
Solve the problem.
54)
Initially, a population of rabbits was found to contain 104 rabbits. It was estimated that the
population was growing exponentially at the rate of 10% per day. Estimate the population after 53
days.
54)
A)
200
B)
150
C)
20,835
D)
1767
55)
A country has a population of 287 million in 2005. Assuming a growth rate of 1.3%, determine the
function that expresses the population of the country t years after 2005.
55)
A)
P(t) = 287e–.013t
B)
P(t) = 287e.013t
C)
P(t) = 287e1.3t
D)
P(t) = 287e–1.3t
E)
none of these
56)
Plutonium has a decay rate of 0.003% per year. What is the half life?
56)
A)
1630 years
B)
2310 years
C)
23,100 years
D)
33,000 years
57)
How long will it take for an investment to triple if interest is paid at 10%, compounded
continuously?
57)
A)
11 years
B)
8.6 years
C)
30 years
D)
3 years
E)
none of these
Solve the problem.
58)
A radioactive substance is observed to disintegrate at a rate such that 9
10 of the original amount
remains after one year. What is the half–life of the substance?
58)
A)
0.301 yr
B)
6.579 yr
C)
0.556 yr
D)
0.588 yr
59)
In a chemical reaction, substance A decomposes at a rate proportional to the amount of A present.
It is found that 8 g of A will reduce to 4 g in 4.2 hours. After how long will there be only 1 g left?
59)
A)
17.1 hours
B)
12.6 hours
C)
9.5 hours
D)
8.4 hours
60)
A savings account paying 3% continuously compounded interest has $1600 after 8 years. How
much was initially deposited?
60)
A)
$1259
B)
$1159
C)
$1020
D)
$630
61)
For the demand function q =p2e–(p + 7), find E(p) and determine if the demand is elastic or
inelastic (or neither) at the price p = 5.
61)
A)
E(p) = p – 2, elastic
B)
E(p) = 2 – p, inelastic
C)
E(p) = 2 – p, elastic
D)
E(p) = p – 2, inelastic
62)
If an investment doubles in 5 years continuously compounded, how long will it take for the
investment to triple?
62)
A)
7.5 years
B)
7.93 years
C)
7 years
D)
10 years
Solve the problem.
63)
Find the doubling time for an amount invested at a growth rate 7% per year compounded
continuously.
63)
A)
9.9 years
B)
7.4 years
C)
4.9 years
D)
8.6 years
64)
How long would it take $6000 to grow to $30,000 at 7% compounded continuously? Round your
answer to the nearest tenth of a year.
64)
A)
23.2 years
B)
14.5 years
C)
23.7 years
D)
23.0 years
65)
What rate of interest is required in order for a $100 investment to double in 6 years if the interest is
compounded continuously?
65)
A)
1.83%
B)
8.7%
C)
11.6%
D)
18.3%
E)
none of these
Solve the problem.
66)
The initial weight of a starving animal is W0. Its weight after t days is given by
W =W0e–0.006t.
What percentage of its initial weight remains after 29 days?
66)
A)
87.4%
B)
16%
C)
79%
D)
84%
67)
$1,000 is invested at r% interest continuously compounded. After 6 years the account has reached
$1350. What was the interest rate?
67)
A)
4.8%
B)
5%
C)
6%
D)
5.3%
Solve the problem.
68)
Find the amount of time required for a $31,000 investment to double if the annual interest rate r is
9.9% and interest is compounded continuously. Round your answer to the nearest hundredth of a
year.
68)
A)
111.46 years
B)
1.04 years
C)
104.46 years
D)
7.00 years
69)
A high–yield savings pays 8% interest, compounded continuously. How long would it take an
initial investment of $2500 to grow to $12,500?
69)
A)
20.118 years
B)
22.397 years
C)
201.18 years
D)
2.01 years
E)
none of these
Solve the problem.
70)
14C has a half life of 5730 years. How old is a piece of charcoal which has lost 90% of its 14C ?
70)
A)
9100 years
B)
20,000 years
C)
11,109 years
D)
19,109 years
E)
none of these
71)
Determine the percentage rate of change of F(t) =e–0.09t2 at t = 1 and t = 5.
71)
A)
F'(1)
F(1) = – 18%, F'(5)
F(5) = – 90%
B)
F'(1)
F(1) = 19%, F'(5)
F(5) = 70%
C)
F'(1)
F(1) = – 19%, F'(5)
F(5) = – 70%
D)
F'(1)
F(1) = 18%, F'(5)
F(5) = 90%
72)
Which of the following functions satisfy the differential equation y’ = 3(12 – y)?
(I) y = 4(3 –e–3x)
(II) y = 12(1 –e–3x)
(III) y = 12 –e–3x
(IV) y = 3(1 –e–12x)
72)
A)
I and III
B)
I and II
C)
II and III
D)
I, II, and III
E)
none of these
73)
For the demand function q = 150(245 –p2), find E(p) and determine if the demand is elastic or
inelastic (or neither) at the price p =7.
73)
A)
E(p) =p2
245 –p2, elastic
B)
E(p) =2p2
245 –p2, inelastic
C)
E(p) =p2
245 –p2, inelastic
D)
E(p) =2p2
245 –p2, elastic
Solve the problem.
74)
An amount is invested at a certain growth rate, k, per year compounded continuously. The
doubling time is 11 years. What is the growth rate k?
74)
A)
7.39%
B)
7.27%
C)
7.62%
D)
6.3%
75)
A certain radioactive element has a half–life of 12 minutes. At what time is the substance decaying
at a rate of 3.466 grams per minute if there are 120 grams present initially?
75)
A)
t = 0.058 min
B)
t = 0
C)
t = 12 min
D)
t = 0.693 min
76)
Radioactive carbon 11 has a half–life of 20 minutes. If there are 200 grams present at the start of our
experiment, how many grams will remain after 10 minutes?
76)
A)
50 g
B)
100 g
C)
6.931 g
D)
141.421 g
77)
Eight years ago, $2000 was deposited in a savings account paying 3% interest compounded
continuously. Three years ago, $500 was withdrawn from the account. What is the current value of
the account?
77)
A)
$2042.50
B)
$3964.93
C)
$1995.41
D)
$4319.08
Solve the problem.
78)
If a population has a growth rate of 6% per year, how long to the nearest tenth of a year will it take
the population to double?
78)
A)
1.2 years
B)
11.6 years
C)
12.3 years
D)
0.1 years
For the demand function given, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price.
79)
q =400 –5p; p =71
79)
A)
E(71) =1
9; inelastic
B)
E(71) =9
71 ; inelastic
C)
E(71) =45; elastic
D)
E(71) =71
9; elastic
Solve the problem.
80)
A beverage company works out a demand function for its sale of soda and finds it to be
q = D(x) =4000 –26x
where q = the quantity of sodas sold when the price per can, in cents, is x. At what price, x, is the
elasticity of demand inelastic?
80)
A)
For x >52,000 cents
B)
For x >308 cents
C)
For x <77 cents
D)
For x <154 cents
81)
A savings account pays 7% interest, compounded continuously. How much should be deposited
now in order to have $5000 in the account at the end of five years?
81)
A)
$533.83
B)
$2884.75
C)
$3523.44
D)
$4661.97
82)
Let y = 6(1 –e–3x). What is 3(6 – y) ?
82)
A)
18e–3x
B)
6 – 6e–3x
C)
3e–3x
D)
12 – 6e–3x
E)
none of these
83)
A bank pays 2.5% interest on deposits. What is the return on a $1000 deposit after two years if
interest is compounded continuously?
83)
A)
$778.80
B)
$1284.03
C)
$1051.27
D)
$1648.72
E)
none of these
Solve the problem.
84)
The size of an insect colony t days after its formation is P( t) = 1000e0.2t. Approximately how many
insects are present after 10 days?
84)
A)
690
B)
7389
C)
54,598
D)
6900
E)
none of these
85)
A certain radioactive substance is decaying at a rate proportional to the amount present. If 100
grams decays to 13.5 grams in 4 years, how long will it take for 90 grams to decay to 30 grams?
85)
A)
0.501 yr
B)
2.195 yr
C)
2.405 yr
D)
Problem cannot be solved as stated.
86)
Initially, a population of rabbits was found to contain 189 rabbits. It was estimated that the
population was growing exponentially at the rate of 11% per day. How long, to the nearest tenth of
a day, will it take the population to double?
86)
A)
0.1 days
B)
16.5 days
C)
63 days
D)
6.3 days
87)
After 5 years of continuous compounding at 12.3% the amount in an account is $11,700. What was
the amount of the initial deposit?
87)
A)
$21,640.97
B)
$6325.50
C)
$5155.50
D)
$20,470.97
For the demand function given, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price.
88)
q =600e–0.04p; p =12
88)
A)
E(12) =0.04; inelastic
B)
E(12) =0.48; inelastic
C)
E(12) =300; elastic
D)
E(12) = 1; neither
Solve the problem.
89)
The population of a colony of bacteria triples in 3 days. Assuming that the rate of growth is
proportional to the size of the population, how long did it take for the colony to double in size?
89)
A)
1.9 days
B)
2 days
C)
6 days
D)
4.9 days
E)
none of these
For the demand function given, find E(p) and determine if demand is elastic or inelastic (or neither) at the indicated price.
90)
q =900 – p; p =59
90)
A)
E(59) =841; elastic
B)
E(59) =59
841; elastic
C)
E(59) =59
841; inelastic
D)
E(59) =1
841; inelastic
Answer Key
Testname: C5
27
Answer Key
Testname: C5
Answer Key
Testname: C5