Question 1

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

Accepted Answer

The answer of Carbon 14 has a half-life of 5730...

Question 2

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

Accepted Answer

The formula for continuous compounding is A = Pe^(rt), where P is the principal, r is the rate, and t is the time. For a $1000 deposit at 2.5% interest for 2 years, A = 1000 * e^(0.025*2) = $1051.27.

Question 3

A fossil was discovered that had about 70% of the ${ } ^ { 14 } \mathrm { C }$ level found today in living matter. Given that the decay constant for ${ } ^ { 14 } \mathrm { C }$ is 0.00012, determine the age of the fossil.
Enter your answer exactly as just $\frac { \ln a } { \mathrm {~b} }$ where a is a real number to one decimal place and b is a real number to 5 decimal places.

Accepted Answer

The answer of A fossil was discovered that had about...

Question 4

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).

Accepted Answer

The answer of Barium 140 has a half-life of 13...

Question 5

Potassium has a half-life of 12 hours. How long will it take for a quantity of potassium to decay to $\frac { 1 } { 10 }$ its original size?
Enter your answer as a real number rounded to one decimal place (no units).

Accepted Answer

The answer of Potassium has a half-life of 12 hours....

Question 6

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

Accepted Answer

The formula for continuously compounded interest is A = Pe^(rt), where P is the principal amount ($500), r is the annual interest rate (0.04), and t is the time in years (32/12). Calculating this gives approximately $556.20.

Question 7

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

Accepted Answer

The formula for continuous compounding is $A = P e^{rt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (decimal), and $t$ is the time the money is invested for in years. To double, $2P = Pe^{r*5}$, solving for $r$ gives $r = \frac{\ln(2)}{5}$. To triple, $3P = Pe^{r*t}$, substituting $r$ and solving for $t$ gives $t = \frac{\ln(3)}{\ln(2)}*5 \approx 7.93$ years.

Question 8

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.

Accepted Answer

The answer of Plutonium 239 has a half-life of 24,000...

Question 9

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

Accepted Answer

We can use the formula for continuous compounding:
A = Pe^(rt)
Where A is the final amount, P is the initial investment, e is the Euler's number (approximately 2.71828), r is the interest rate, and t is the time in years.
Plugging in the given values, we get:
$20,000 = $2500e^(0.2t)
Dividing both sides by $2500 and taking the natural logarithm of both sides, we get:
ln(8) = 0.2t
t = ln(8)/0.2
t ≈ 10.397 years
Therefore, the correct answer is C: 10.397 years.

Question 10

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

Accepted Answer

To find the time it takes for an investment to triple with continuous compounding, use the formula $ t = \frac{\ln(3)}{r} $. With $ r = 0.10 $, $ t = \frac{\ln(3)}{0.10} \approx 10.986 $ years, which is approximately 11 years.

Question 11

$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%

Accepted Answer

The answer of $1,000 is invested at r% interest continuously...

Question 12

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).

Accepted Answer

The answer of Krypton 85 gas leaks into the reactor...

Question 13

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

Accepted Answer

The formula for continuous compounding is A = Pe^(rt). To double the investment, A = 2P. Solving 2 = e^(3r) gives r ≈ 0.231 or 23.1%.

Question 14

$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

Accepted Answer

The formula for continuous compounding is given as A = Pe^(rt), where A is the ending amount, P is the principal amount, e is the mathematical constant (approx. 2.71828), r is the annual interest rate as a decimal, and t is the time in years. Plugging in the given values, we get:

A = 1000*e^(0.06*5)
A = $1349.86

Therefore, the value of the investment after 5 years is $1349.86, which corresponds to option A.

Question 15

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

Accepted Answer

The formula for continuously compounded interest is $A = Pe^{rt}$, where $A$ is the amount of money accumulated after n years, including interest, $P$ is the principal amount (the initial amount of money), $r$ is the annual interest rate (decimal), and $t$ is the time the money is deposited for in years. Rearranging the formula to solve for $P$, we get $P = \frac{A}{e^{rt}}$. Plugging in the given values ($A = 1600$, $r = 0.03$, $t = 8$), we find $P = \frac{1600}{e^{0.03 \times 8}}$, which calculates to approximately $1259.

Question 16

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 $\mathrm { P } _ { 0 }$ of strontium 90 to decay to $\frac { 1 } { 3 }$ its original size?
Enter your answer exactly in the form $\frac { \ln a } { \mathrm {~b} }$ where a is a reduced fraction and b is a real number to four decimal places (no units).

Accepted Answer

The answer of The decay constant for strontium 90 is...

Question 17

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

Accepted Answer

The formula for continuous compounding is A=Pe^rt, where A is the final amount, P is the principal (initial amount), r is the annual interest rate in decimal form, and t is the time in years. We are given A=$5000, r=0.07, and t=5. Solving for P, we get P=A/e^rt=$3523.44. Therefore, the best choice is C.

Question 18

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

Accepted Answer

The answer of Eight years ago, $2000 was deposited in...

Question 19

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

Accepted Answer

The answer of How much money has to be invested...

Question 20

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).

Accepted Answer

The answer of It is observed that the sales of...

Quiz 5: Applications of the Exponential and Natural Logarithm Functions

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

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

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).

Potassium has a half-life of 12 hours. How long will it take for a quantity of potassium to decay to 110\frac { 1 } { 10 }101​ its original size? Enter your answer as a real number rounded to one decimal place (no units).

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

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

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.

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

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

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

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).

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

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

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

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?

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?

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

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).

Potassium has a half-life of 12 hours. How long will it take for a quantity of potassium to decay to $\frac { 1 } { 10 }$ its original size? Enter your answer as a real number rounded to one decimal place (no units).