Deck 5: Applications of the Exponential and Natural Logarithm Functions

Question 1

${ } ^ { 14 } \mathrm { C }$ has a half life of 5730 years. How old is a piece of charcoal which has lost 90% of its ${ } ^ { 14 } \mathrm { C }$ ?

A) 20,000 years
B) 11,109 years
C) 9100 years
D) 19,109 years
E) none of these

Answer 1

19,109 years

Question 2

A radioactive substance is observed to disintegrate at a rate such that $\frac { 9 } { 10 }$ of the original amount remains after one year. What is the half-life of the substance?

A) 0.301 yr
B) 0.588 yr
C) 0.556 yr
D) 6.579 yr

Answer 2

6.579 yr

Question 3

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.

A) P(t) = 287 $e ^ { - .013t }$
B) P(t) = 287 $e ^ { .013t }$
C) P(t) = 287 $e ^ { 1.3 t }$
D) P(t) = 287 $e ^ { - 1.3t }$
E) none of these

Answer 3

P(t) = 287 $e ^ { .013t }$

Question 4

Plutonium has a decay rate of 0.003% per year. What is the half life?

A) 2310 years
B) 23,100 years
C) 33,000 years
D) 1630 years

Question 5

Assume that a culture of bacteria grows at a rate proportional to its size such that if bacteria are present initially, then there are 2 × bacteria present after 3 hours. Determine a formula for the number of bacteria present after t hours in terms of powers of 2. (Hint: Recall that = .)Enter your answer exactly in the form

Question 6

In a certain country, the rate of increase of the population is proportional to the population P(t). In fact, $P ^ { \prime } ( t ) = 0.23 P ( t )$ Suppose that initially the country's population is 50,000, and that 10 years later there are 500,000 people. Which of the following equations expresses this information mathematically?

A) 10 = $e ^ { 2.3 t }$
B) 500 = $e ^ { 2.3 ( 50 ) }$
C) 500,000 = $\mathrm { e } ^ { 0.23 ( 10 ) }$
D) 500,000 = 50,000 $e ^ { 2.3 }$
E) none of these

Question 7

The size of an insect colony t days after its formation is P( t) = 1000 $e ^ { 0.2 t }$ . Approximately how many insects are present after 10 days?

A) 690
B) 7389
C) 54,598
D) 6900
E) none of these

Question 8

A bacterial culture grows exponentially; that is, P(t) = 100 $\mathrm { e^{kt} }$ , 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)?

A) 3
B) 0.69
C) 0.06
D) 0.48
E) none of these

Question 9

Suppose that a school of fish in a pond grows according to the exponential law and suppose that the size of the colony triples in 24 days. Determine k.
Enter your answer exactly in the form where a, b are integers.

Question 10

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?

A) 141.421 g
B) 6.931 g
C) 50 g
D) 100 g

Question 11

A parchment is offered for sale at a Paris flea market. The owner claims it is at least 2000 years old. However, a carbon-dating test shows that - ratio for the manuscript is 95% of the corresponding ratio for currently manufactured parchment. How old is the manuscript? (The decay constant of is 0.00012)Enter your answer exactly in the form (no units).

Question 12

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?

A) 2.405 yr
B) 2.195 yr
C) 0.501 yr
D) Problem cannot be solved as stated.

Question 13

A colony of bacteria is growing at a rate proportional to the number of bacteria present. At the beginning of an experiment there were about bacteria present. In two hours, the count rose to bacteria. At what time will there be 6 × bacteria present?
Enter just a real number rounded up to one decimal place (no units).

Question 14

Suppose that at any time t, a colony of fruit flies is growing at a rate equal to one half the current size of the colony. Find a formula which gives the size of the colony at time t if there were originally 500 fruit flies present.
Enter your answer exactly in the form:

Question 15

Suppose that a school of fish in a pond grows according to the exponential law and suppose that the size of the colony triples in 24 days. If the initial size of the school was 50, when will the school contain 200 fish? Enter your answer exactly in the form

Question 16

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?

A) t = 0
B) t = 0.693 min
C) t = 0.058 min
D) t = 12 min

Question 17

Let P(t) be the quantity of strontium-90 remaining after t years. Suppose the half-life of strontium-90 is 28 years. Which of the following equations expresses the half-life information?

A) P(28) = $\frac { 1 } { 2 }$ $\mathrm { P } _ { 0 }$
B) P $\frac { 1 } { 2 }$ = 28 $\mathrm { P } _ { 0 }$
C) 28 = $\frac { 1 } { 2 }$ $\mathrm { P } _ { 0 }$
D) 28 = $\frac { 1 } { 2 }$ $\mathrm { P } _ { 0 }$ $\mathrm { e } ^ { - \mathrm { kt } }$
E) none of these

Question 18

The population of a certain region was 10 million in 1950. By 1970, it had increased to 13.5 million. Assuming exponential growth, estimate the population in the year 2000.
Enter your answer exactly in the form a (no units).

Question 19

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?

A) 2 days
B) 1.9 days
C) 4.9 days
D) 6 days
E) none of these

Question 20

Sixteen pounds of a radioactive substance loses one fourth of its original mass in 2 days. The mass m(t) remaining at time t is given by:

A) m(t) = 16 $e ^ { - 0.693t }$
B) m(t) = 16 $e ^ { 0.693t }$
C) m(t) = 4 $e ^ { - 0.693t }$
D) m(t) = 16 $e ^ { 0.144t }$

Question 21

Carbon 14 has a half-life of 5730 years. Determine its decay constant.
Enter just a real number rounded to five decimal places.

Question 22

A bank pays 2.5% interest on deposits. What is the return on a $1000 deposit after two years if interest is compounded continuously?

A) $1051.27
B) $778.80
C) $1648.72
D) $1284.03
E) none of these

Question 23

A fossil was discovered that had about 70% of the level found today in living matter. Given that the decay constant for is 0.00012, determine the age of the fossil.
Enter your answer exactly as just where a is a real number to one decimal place and b is a real number to 5 decimal places.

Question 24

Barium 140 has a half-life of 13 days. After 25 days a given sample comprises 5 grams. How large was the original sample?
Enter your answer as a real number to two decimal places (no units).

Question 25

Potassium has a half-life of 12 hours. How long will it take for a quantity of potassium to decay to its original size?
Enter your answer as a real number rounded to one decimal place (no units).

Question 26

You deposit $500 in a savings account paying 4% interest continuously compounded. How much will the account be worth after 32 months?

A) $545.20
B) $526.10
C) $576.70
D) $556.20

Question 27

If an investment doubles in 5 years continuously compounded, how long will it take for the investment to triple?

A) 7.93 years
B) 7 years
C) 10 years
D) 7.5 years

Question 28

Plutonium 239 has a half-life of 24,000 years. What is its decay constant?
Enter your answer as just a real number to eight decimal places.

Question 29

A high-yield savings pays 20% interest, compounded continuously. How long would it take an initial investment of $2500 to grow to $20,000?

A) 1.04 years
B) 103.972 years
C) 10.397 years
D) 14.979 years
E) none of these

Question 30

How long will it take for an investment to triple if interest is paid at 10%, compounded continuously?

A) 30 years
B) 3 years
C) 8.6 years
D) 11 years
E) none of these

Question 31

$1,000 is invested at r% interest continuously compounded. After 6 years the account has reached $1350. What was the interest rate?

A) 6%
B) 5.3%
C) 4.8%
D) 5%

Question 32

Krypton 85 gas leaks into the reactor room of an electric power plant. Its half-life is 10 years. How long is it before 99.9% of the krypton decays?
Enter your answer as just a real number rounded to one decimal place (no units).

Question 33

What rate of interest is required in order for a $100 investment to double in 3 years if the interest is compounded continuously?

A) 23.1%
B) 3.05%
C) 5.49%
D) 54.9%
E) none of these

Question 34

$1000 is invested at 6% interest compounded continuously. What is the value of the investment after 5 years?

A) $1349.86
B) $6691.13
C) $1338.23
D) $1822.12
E) none of these

Question 35

A savings account paying 3% continuously compounded interest has $1600 after 8 years. How much was initially deposited?

A) $1020
B) $1159
C) $1259
D) $630

Question 36

The decay constant for strontium 90 is λ = 0.0244, where the time is measured in years. How long will it take for a quantity of strontium 90 to decay to its original size?
Enter your answer exactly in the form where a is a reduced fraction and b is a real number to four decimal places (no units).

Question 37

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?

A) $4661.97
B) $2884.75
C) $3523.44
D) $533.83

Question 38

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?

A) $4319.08
B) $2042.50
C) $3964.93
D) $1995.41

Question 39

How much money has to be invested now at 8% continuous interest in order to have $1000 after 5 years?

A) $461.56
B) $18.32
C) $183.20
D) $670.32
E) none of these

Question 40

It is observed that the sales of a certain recording fall to 75% of their original level one month after advertising stops. If this continues, what will be the sales after 4 months?
Enter just a real number rounded to one decimal place (no units or symbols).

Question 41

Suppose a manufacturer can sell q = - 6 units of a product when the price is p dollars per unit. Determine the elasticity of demand, E(p), when the price is p = 8 dollars.
Enter just an integer.

Question 42

Let y = 6(1 - $e ^ { - 3 x }$ ). What is 3(6 - y) ?

A) 12 - 6 $e ^ { - 3 x }$
B) 18 $e ^ { - 3 x }$
C) 6 - 6 $e ^ { - 3 x }$
D) 3 $e ^ { - 3 x }$
E) none of these

Question 43

Suppose that $500 is deposited in a bank certificate paying 10% interest compounded continuously. How much will the certificate be worth after 5 years? Enter your answer in dollars to two decimal places (no symbols or words).

Question 44

Determine the percentage rate of change of f(x) = $e ^ { 0.9 x }$ at x = 15 and x = 30.

A) $\frac { f ^ { \prime } ( 15 ) } { f ( 15 ) }$ = 22%, $\frac { f ^ { \prime } ( 30 ) } { f ( 30 ) }$ = 5%
B) $\frac { f ^ { \prime } ( 15 ) } { f ( 15 ) }$ = 90%, $\frac { f ^ { \prime } ( 30 ) } { f ( 30 ) }$ = 90%
C) $\frac { f ^ { \prime } ( 15 ) } { f ( 15 ) }$ = 38%, $\frac { f ^ { \prime } ( 30 ) } { f ( 30 ) }$ = 14%
D) $\frac { f ^ { \prime } ( 15 ) } { f ( 15 ) }$ = 17%, $\frac { f ^ { \prime } ( 30 ) } { f ( 30 ) }$ = 29%

Question 45

Suppose that the value of a certain investment after t years can be approximated by the function What is the dollar value of the investment after 8 years?
Enter your answer exactly in the form: a where a is an integer and b is a real number to two decimal places (no units).

Question 46

A manufacturer has a demand function q = . Determine the elasticity of demand as a function of p.
Enter your answer exactly as where P, Q are polynomials in p in standard form, and the leading coefficient of P is a reduced fraction of form .

Question 47

For the demand function q = $p ^ { 2 }$ $e ^ { - ( p + 7 ) }$ , find E(p) and determine if the demand is elastic or inelastic (or neither) at the price p = 5.

A) E(p) = 2 - p, inelastic
B) E(p) = p - 2, elastic
C) E(p) = p - 2, inelastic
D) E(p) = 2 - p, elastic

Question 48

Mr. Jones has two investments. The first is currently worth $50,000 and has an annual yield of 10% compounded continuously. The second is currently worth $70,000 and has an annual yield of 8% compounded continuously. After how many years will the two investments be worth the same amount? Enter your answer in years to one decimal place (no units).

Question 49

Determine the percentage rate of change of F(t) = $\mathrm { e } ^ { - 0.09 \mathrm { t } ^ { 2 } }$ at t = 1 and t = 5.

A) $\frac { F ^ { \prime } ( 1 ) } { F ( 1 ) }$ = 18%, $\frac { F ^ { \prime } ( 5 ) } { F ( 5 ) }$ = 90%
B) $\frac { F ^ { \prime } ( 1 ) } { F ( 1 ) }$ = -18%, $\frac { F ^ { \prime } ( 5 ) } { F ( 5 ) }$ = -90%
C) $\frac { F ^ { \prime } ( 1 ) } { F ( 1 ) }$ = 19%, $\frac { F ^ { \prime } ( 5 ) } { F ( 5 ) }$ = 70%
D) $\frac { F ^ { \prime } ( 1 ) } { F ( 1 ) } = - 19 \% , \frac { F ^ { \prime } ( 5 ) } { F ( 5 ) } = - 70 \%$

Question 50

How long is required for an investment of $2000 to double if the interest is 10% compounded continuously? Enter your answer in years to one decimal place

Question 51

A high school student deposits a $500 graduation gift in a bank account that pays 4.8% interest compounded continuously. How much will the account be worth after 18 months?

A) $1123.95
B) $1027.22
C) $973.71
D) $1162.71
E) none of these

Question 52

Determine the percentage rate of change of h(p) = $\frac { 4 } { 7 p + 3 }$ at p = 1 and p = 9.

A) $\frac { h ^ { \prime } ( 1 ) } { h ( 1 ) }$ = 40%, $\frac { h ^ { \prime } ( 9 ) } { h ( 9 ) }$ = 6%
B) $\frac { h ^ { \prime } ( 1 ) } { h ( 1 ) }$ = -70%, $\frac { h ^ { \prime } ( 9 ) } { h ( 9 ) }$ = -11%
C) $$\frac { h ^ { \prime } ( 1 ) } { h ( 1 ) }$$ = -40%, $\frac { h ^ { \prime } ( 9 ) } { h ( 9 ) }$ = -6%
D) $\frac { h ^ { \prime } ( 1 ) } { h ( 1 ) }$ = 70%, $\frac { h ^ { \prime } ( 9 ) } { h ( 9 ) }$ = 11%

Question 53

For the demand function q = 150(245 - $p ^ { 2 }$ ), find E(p) and determine if the demand is elastic or inelastic (or neither) at the price p =7.

A) E(p) = $\frac { p ^ { 2 } } { 245 - p ^ { 2 } }$ , elastic
B) E(p) = $\frac { 2 p ^ { 2 } } { 245 - p ^ { 2 } }$ , elastic
C) E(p) = $\frac { p ^ { 2 } } { 245 - p ^ { 2 } }$ , inelastic
D) E(p) = $\frac { 2 p ^ { 2 } } { 245 - p ^ { 2 } }$ , inelastic

Question 54

Determine the percentage rate of change of f(t) = 2 at t = 5.
Enter your answer as just a reduced fraction of form .

Question 55

What is the present value of an investment of $1000 payable at the end of 10 years at a 9% rate of interest compounded continuously? Enter your answer in dollars to two decimal places (no symbols or words).

Question 56

Suppose that the value in billions of dollars of a company is determined to be where t is measured in years. What is the percentage rate of growth of the company at time t = 0?
Enter just an integer (no units).

Question 57

Which of the following functions satisfy the differential equation y' = 3(12 - y)?
(I) y = 4(3 - $e ^ { - 3 x }$ )
(II) y = 12(1 - $e ^ { - 3 x }$ )
(III) y = 12 - $e ^ { - 3 x }$
(IV) y = 3(1 - $e ^ { - 12 x }$ )

A) I, II, and III
B) I and II
C) I and III
D) II and III
E) none of these

Question 58

What rate of interest will make an investment triple in 8 years if the interest is compounded continuously? Enter your answer as a percent to one decimal place

Question 59

A company has a demand function q= 200 - 40p. Determine the elasticity of demand as a function of p. Enter your answer exactly as where P, Q are polynomials in p in standard form.

Question 60

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?

A) f'(t) + $\frac { 1 } { 10 }$ f(t) = 10,000
B) f(t) = $\frac { 10 f ^ { \prime } ( t ) } { 10,000 }$
C) f'(t) = $\frac { 1 } { 10 }$ (10,000 - f(t))
D) f'(t) = $\frac { 1 } { 10 }$ f(t)
E) none of these

Question 61

A rumor is spread by word of mouth to a potential audience of 3000 people. After t days, people will have heard the rumor. The graphs are shown below.


At what rate will the rumor be spreading when half of the potential audience has heard the rumor? Enter just an integer rounded to the nearest hundred (no units).

Question 62

A rumor is spread by word of mouth to a potential audience of 3000 people. After t days, people will have heard the rumor. The graphs are shown below.


Approximately when will 1250 people have heard the rumor? Enter just an integer (no units).

Question 63

A rumor is spread by word of mouth to a potential audience of 3000 people. After t days, people will have heard the rumor. The graphs are shown below.


Approximately when will the rumor be spreading at a rate of 200 people per day? Enter just two integers a, b where a< (round to the nearest integers).

Question 64

A rumor is spread by word of mouth to a potential audience of 3000 people. After t days, people will have heard the rumor. The graphs are shown below.


At approximately what rate will the rumor be spreading after 7 days? Enter just an integer rounded to the nearest hundred (no words or units).

Question 65

If the relative rate of change of a function f is always 5, what kind of equation will f be represented by? Enter your answer in the form: f(t) = C (leave C in your answer).

Question 66

If y = 10(1 - ) and y' = a -2y, find a.
Enter just an integer.

Question 67

A rumor is spread by word of mouth to a potential audience of 3000 people. After t days, people will have heard the rumor. The graphs are shown below.


Approximately how many people will have heard the rumor after 7 days? Enter your answer as just an integer rounded to the nearest hundred.

Question 68

A rumor is spread by word of mouth to a potential audience of 3000 people. After t days, people will have heard the rumor. The graphs are shown below.


When will the rumor be spreading at the greatest rate? Enter just an integer (round to the nearest integer, no units)

Question 69

Let g(t) = 100 - 100 $e ^ { - 0.01 t }$ be the number of cases of measles in a certain school t days after the first case is reported. Which of the following best describes the spread of the disease?

A) The number of cases decreases according to exponential decay.
B) After the first day there are 100 cases, after which the number of cases decreases daily to 0.
C) Initially, the disease spreads quickly, but then the rate of increase slows so that the number of cases never exceeds 100.
D) The disease spreads quickly until the number of cases reaches 100; then gradually the number of cases decreases.
E) none of these

Question 70

If y = 2(1 - ) and y' = 4 - by, find b.
Enter just an integer.

Question 71

Let A'(t) = 0.3(25,000 - A(t)), A(0) = 0. Which of the following is the formula for A(t)?

A) A(t) = 25,000(1 - $e ^ { - 0.3 t}$ )
B) A(t) = 25,000 $e ^ { - 3 t }$
C) A(t) = 25,000(1 - 0.3t)
D) A(t) = 25,000 $e ^ { ( 1 - 0.3 t ) }$
E) none of these

Question 72

Find a so that y = 1 - a solves the differential equation Enter just an integer.

Question 73

Find a so that y = 1 + solves the differential equation Enter just a reduced fraction of form .