Question 1

How much would an initial amount of $4,000, compounded continuously at 5% annual interest, become after 10 years?

Accepted Answer

$6,594.89

Question 2

How much would an initial amount of $5,000, compounded continuously at 8% annual interest, become after 28 years?

Accepted Answer

$46,966.66

Question 3

How much would you need to deposit in a bank account paying 4% annual interest compounded continuously so that at the end of 2 years you would have $10,000?
A) $9,231.16
B) $9,607.89
C) $10,832.87
D) $10,408.11

Accepted Answer

To find the initial deposit required to grow to $10,000 in 2 years at a 4% annual interest rate compounded continuously, you use the formula for continuous compounding: $A = P e^{rt}$, where $A$ is the amount after time $t$, $P$ is the principal amount (initial deposit), $r$ is the annual interest rate (as a decimal), and $e$ is the base of the natural logarithm (approximately 2.71828). Rearranging the formula to solve for $P$, we get $P = \frac{A}{e^{rt}}$. Plugging in the values ($A = 10,000$, $r = 0.04$, $t = 2$), we get $P = \frac{10,000}{e^{0.04 \times 2}}$, which calculates to approximately $9,231.16.

Question 4

How much would you need to deposit in a bank account paying 4% annual interest compounded continuously so that at the end of 19 years you would have $20,000?

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Question 5

Suppose that a bank account that compounds interest continuously grows from $1000 to $1100 in 5 years. What annual interest rate is the bank paying? Round your answer to one decimal place.

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Question 6

Suppose that a bank account that compounds interest continuously grows from $200 to $310 in three years. What annual interest rate is the bank paying? Round your answer to one decimal place.

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Question 7

Suppose a colony of bacteria has a continuous growth rate of 18% per hour. By what percent will the colony have grown after 5 hours?
A) 40.7%
B) 246.0%
C) 60.7%
D) 146.0%

Accepted Answer

The growth of the bacteria can be calculated using the formula for continuous growth: $A = P e^{rt}$, where $A$ is the amount after time $t$, $P$ is the initial amount, $e$ is the base of the natural logarithm (approximately 2.71828), $r$ is the rate of growth, and $t$ is the time in the same units as the rate. For an 18% growth rate per hour over 5 hours, the formula becomes $A = P e^{0.18 \times 5}$. Calculating this gives a growth factor of approximately 2.46, meaning the colony has grown 146% beyond its original size.

Question 8

Suppose a colony of bacteria has a continuous growth rate of 20% per hour. By what percent will the colony have grown after 2 hours? Round your answer to the nearest tenth.

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Question 9

Suppose a country's population increases by a total of 5% over a 2-year period. What is the continuous growth rate for this country? Round your answer to the nearest hundredth.

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Question 10

Suppose a country's population increases by a total of 17% over a 7-year period. What is the continuous growth rate for this country? Round your answer to the nearest hundredth.

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Question 11

Suppose that the amount of the world's computer hard disk storage increases by a total of 300% over a 9-year period. What is the continuous growth rate for the amount of the world's hard disk storage?
A) 7.7%
B) 0.9%
C) 15.4%
D) 1.7%

Accepted Answer

The continuous growth rate is given by the formula r = ln(1 + g/100) / t, where g is the growth rate in percentage and t is the time in years. In this case, g = 300% and t = 9 years, so r = ln(1 + 300/100) / 9 = ln(4) / 9 = 0.154 = 15.4%.

Question 12

Suppose that a colony of bacteria has a continuous growth rate of 28% per hour. If the colony contains 10,000 cells now, how many did it contain 8 hours ago?

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Question 13

Suppose that a colony of bacteria has a continuous growth rate of 33% per hour. How many hours does it take the colony to triple in size? Round your answer to the nearest hundredth.

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Question 14

About how many years does it take for money to double when compounded continuously at 4% per year?
A) 35
B) 9
C) 18
D) 17

Accepted Answer

The formula for continuous compounding is $A = P e^{rt}$, where $A$ is the amount of money after time $t$, $P$ is the principal amount, $r$ is the annual interest rate, and $e$ is the base of the natural logarithm. To double the money, $A/P = 2$, so $2 = e^{0.04t}$. Taking the natural logarithm of both sides gives $ln(2) = 0.04t$, and solving for $t$ gives $t = \frac{ln(2)}{0.04} \approx 17.33$ years, which rounds to 17 years.

Question 15

About how many years does it take for $200 to become $800 when compounded continuously at 5% per year?

Accepted Answer

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Question 16

How many years does it take for money to triple when compounded continuously at 9% per year? Round your answer to the nearest year.

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Question 17

Find a formula for estimating how long money takes to increase by a factor of 9 at R percent annual interest compounded continuously.
A)
 
B)
 
C)
 
D)

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Question 18

Suppose one bank account pays 8% annual interest compounded once per year, and a second bank account pays 8% annual interest compounded continuously. If both bank accounts start with the same initial amount, how many years will it take for the second bank account to contain double the amount of the first bank account?

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Question 19

Suppose one bank account pays 9% annual interest compounded once per year, and a second bank account pays 10% annual interest compounded continuously. If both bank accounts start with the same initial amount, how many years will it take for the second bank account to contain 50% more than the first bank account?

Accepted Answer

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Question 20

Suppose a colony of bacteria has doubled in 9 hours. What is the approximate percent per hour continuous growth rate of this colony of bacteria? Round your answer to the nearest whole percent.

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Quiz 4: Exponential Functions, Logarithms, and E