Question 1

Solve the problem. Round your answer to three decimals.
-What annual rate of interest is required to double an investment in 10 years?
A) 6.931%
B) 11.612%
C) 7.177%
D) 3.589%

Accepted Answer

The rule of 72 states that to find the number of years it takes for an investment to double, divide 72 by the annual rate of interest. Therefore, to double an investment in 10 years, the annual rate of interest should be 7.2% (72/10). Since answer choice C, 7.177%, is the closest value to 7.2%, it is the best choice.

Question 2

Solve the problem.
-Larry has $2,000 to invest and needs $2,500 in 13 years. What annual rate of return will he need to get in order to accomplish his goal? (Round your answer to two decimals.)
A) 3.45%
B) 1.45%
C) 2.45%
D) 1.72%

Accepted Answer

The answer of Solve the problem.
-Larry has $2,000 to invest...

Question 3

Solve the problem.
-Conservationists tagged 130 black-nosed rabbits in a national forest in 1990. In 1,991, they tagged 260 black-nosed rabbits in the same range. If the rabbit population follows the exponential law, how many rabbits
Will be in the range 9 years from 1990?
A) 66,560
B) 281
C) 133,120
D) 140

Accepted Answer

The answer of Solve the problem.
-Conservationists tagged 130 black-nosed rabbits...

Question 4

Solve the problem. Round your answer to three decimals.
-What annual rate of interest is required to triple an investment in 7 years?
A) 10.409%
B) 16.993%
C) 8.497%
D) 15.694%

Accepted Answer

The answer of Solve the problem. Round your answer to...

Question 5

Solve the problem.
-Julio figures that he can save $5000 per year. If, at the end of each year, he invests the money in a certificate of
deposit (CD) which pays 7% interest annually, how much money will he have saved in six years?

Accepted Answer

The answer of Solve the problem.
-Julio figures that he can...

Question 6

Solve the problem.
-The size P of a small herbivore population at time t (in years) obeys the function $\mathrm { P } ( \mathrm { t } ) = 800 \mathrm { e } ^ { 0.24 \mathrm { t } }$ if they have enough food and the predator population stays constant. After how many years will the population reach 1,600?
A) 8.66 yrs
B) 27.85 yrs
C) 7.05 yrs
D) 2.89 yrs

Accepted Answer

To find the time t when the population reaches 1,600, solve the equation $800 \mathrm{e}^{0.24t} = 1600$. Dividing both sides by 800 gives $\mathrm{e}^{0.24t} = 2$. Taking the natural logarithm of both sides, we get $0.24t = \ln(2)$. Solving for t gives $t = \frac{\ln(2)}{0.24} \approx 2.89$ years.

Question 7

Solve the problem.
-How long does it take $1125 to triple if it is invested at 7% interest, compounded quarterly? Round your answer to the nearest tenth.
A) 15.8 years
B) 18.1 months
C) 15.8 months
D) 18.1 years

Accepted Answer

The answer of Solve the problem.
-How long does it take...

Question 8

Solve the problem.
-How long does it take $1700 to double if it is invested at 5% interest, compounded monthly? Round your
answer to the nearest tenth.

Accepted Answer

The answer of Solve the problem.
-How long does it take...

Question 9

Solve the problem.
-The Feldmans bought their first house for $14,000. Over the years they moved three times into bigger and bigger houses. Now, 41 years later, they are ready to retire and want a smaller house like the first one they
Bought. If inflation in property values has averaged 3.6% per year during that time, how much will such a
House cost them now? (Round your answer to the nearest dollar.)
A) $3,284
B) $3,200
C) $61,256
D) $59,687

Accepted Answer

The answer of Solve the problem.
-The Feldmans bought their first...

Question 10

Solve the problem.
-If Emery has $1,100 to invest at 9% per year compounded monthly, how long will it be before he has $1,400? If the compounding is continuous, how long will it be? (Round your answers to three decimal places.)
A) 0.036 yrs, 0.268 yrs
B) 66.89 yrs, 2.902 yrs
C) 2.69 yrs, 2.68 yrs
D) 0.233 yrs, 0.223 yrs

Accepted Answer

For monthly compounding, the formula used is $A = P(1 + \frac{r}{n})^{nt}$, 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), $n$ is the number of times that interest is compounded per year, and $t$ is the time the money is invested for in years. For continuous compounding, the formula is $A = Pe^{rt}$, where $e$ is the base of the natural logarithm, approximately equal to 2.71828. Given: $P = 1100$, $A = 1400$, $r = 0.09$, and for monthly compounding, $n = 12$.1. For monthly compounding:$$1400 = 1100(1 + \frac{0.09}{12})^{12t}$$Solving for $t$ gives approximately $t = 2.69$ years.2. For continuous compounding:$$1400 = 1100e^{0.09t}$$Solving for $t$ gives approximately $t = 2.68$ years.Therefore, it will take approximately 2.69 years with monthly compounding and 2.68 years with continuous compounding for Emery to have $1,400.

Question 11

Solve the problem.
-A venture capital firm invested $2,000,000 in a new company in 1995. In 1999, they sold their stake in the
company for $10,500,000. What was the average annual rate of return on their investment?

Accepted Answer

The answer of Solve the problem.
-A venture capital firm invested...

Question 12

Solve the problem.
-The revenue for a dot.com company is projected to double each year for the first 5 years. If the revenue for the
first year is $2 million, write a function showing the revenue R after x years. What is the revenue for the fourth
year?

Accepted Answer

The answer of Solve the problem.
-The revenue for a dot.com...

Question 13

Solve the problem.
-Tracey bought a diamond ring appraised at $1,700 at an antique store. If diamonds have appreciated in value at an annual rate of 10%, what was the value of the ring 10 years ago? (Round your answer to the nearest dollar.)
A) $230
B) $655
C) $625
D) $4,409

Accepted Answer

To find the value of the ring 10 years ago, we use the formula for exponential decay, which is the inverse of exponential growth: $P = P_0(1 + r)^{-t}$, where $P$ is the future value, $P_0$ is the initial value, $r$ is the rate of appreciation, and $t$ is the time in years. Rearranging to solve for $P_0$, we get $P_0 = P / (1 + r)^t$. Plugging in the values: $P_0 = 1700 / (1 + 0.10)^{10}$, which calculates to approximately $655 when rounded to the nearest dollar.

Question 14

Solve the problem.
-Gillian has $10,000 to invest in a mutual fund. The average annual rate of return for the past five years was
12.25%. Assuming this rate, determine how long it will take for her investment to double.

Accepted Answer

The answer of Solve the problem.
-Gillian has $10,000 to invest...

Question 15

Solve the problem. Round your answer to three decimals.
-How long will it take for an investment to double in value if it earns 9.25% compounded continuously?
A) 7.493 years
B) 3.747 years
C) 7.781 years
D) 11.877 years

Accepted Answer

We can use the formula A=P*e^(rt) where A is the final amount, P is the initial amount, r is the annual interest rate, and t is the time (in years). We want to solve for t when A=2P and r=0.0925. Plugging these values into the formula and simplifying, we get:

2 = e^(0.0925t)
ln(2) = 0.0925t
t = ln(2)/0.0925

Using a calculator, we get t ≈ 7.493 years. Rounding to three decimals, the answer is 7.493 years, which corresponds to choice A.

Question 16

Solve the problem. Round your answer to three decimals.
-How long will it take for an investment to triple in value if it earns 8.5% compounded continuously?
A) 12.925 years
B) 8.155 years
C) 6.462 years
D) 13.797 years

Accepted Answer

The formula for continuous compounding 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 invested for in years. To triple an investment, $A/P = 3$. Substituting the given values and solving for $t$, we get $3 = e^{0.085t}$. Taking the natural logarithm of both sides gives $ln(3) = 0.085t$. Solving for $t$ gives $t = ln(3)/0.085 \approx 12.925$ years.

Question 17

Solve the problem.
-The bacteria in a container quadruples every day. If there are initially 100 bacteria, write an equation that
models the number of bacteria A after d days. How many bacteria will there be after 1 week?

Accepted Answer

The answer of Solve the problem.
-The bacteria in a container...

Question 18

Solve the problem.
-How much money needs to be invested now to get $2000 after 4 years at 8% compounded quarterly? Express your answer to the nearest dollar.
A) $2746
B) $1848
C) $1457
D) $584

Accepted Answer

The answer of Solve the problem.
-How much money needs to...

Question 19

Solve the problem.
-In a networking marketing plan for a company, each distributor is expected to recruit 3 new distributors. Jack
was the first distributor hired by the company, so he is considered a level 1 distributor. The 3 people he recruits
are considered level 2 distributors. The people recruited by the level 2 distributors are considered level 3
distributors, and so on. Write a function that models the number of distributors D at each level L. How many
distributors would there be at the fifth level?

Accepted Answer

The answer of Solve the problem.
-In a networking marketing plan...

Question 20

Solve the problem.
-Cindy will require $12,000 in 5 years to return to college to get an MBA degree. How much money should she ask her parents for now so that, if she invests it at 11% compounded continuously, she will have enough for
School? (Round your answer to the nearest dollar.)
A) $3,994
B) $7,121
C) $6,923
D) $20,799

Accepted Answer

We can use the formula A = Pe^rt, where A is the amount after time t, P is the initial investment, r is the interest rate, and e is the mathematical constant. Plugging in the values, we get 12000 = P*e^(0.11*5), which simplifies to P = 6923. Therefore, Cindy should ask her parents for $6,923.

Quiz 4: Exponential and Logarithmic Functions

Solve the problem. Round your answer to three decimals. -What annual rate of interest is required to double an investment in 10 years?

Solve the problem. -Larry has $2,000 to invest and needs $2,500 in 13 years. What annual rate of return will he need to get in order to accomplish his goal? (Round your answer to two decimals.)

Solve the problem. -Conservationists tagged 130 black-nosed rabbits in a national forest in 1990. In 1,991, they tagged 260 black-nosed rabbits in the same range. If the rabbit population follows the exponential law, how many rabbits Will be in the range 9 years from 1990?

Solve the problem. Round your answer to three decimals. -What annual rate of interest is required to triple an investment in 7 years?

Solve the problem. -Julio figures that he can save $5000 per year. If, at the end of each year, he invests the money in a certificate of deposit (CD) which pays 7% interest annually, how much money will he have saved in six years?

Solve the problem. -How long does it take $1125 to triple if it is invested at 7% interest, compounded quarterly? Round your answer to the nearest tenth.

Solve the problem. -How long does it take $1700 to double if it is invested at 5% interest, compounded monthly? Round your answer to the nearest tenth.

Solve the problem. -If Emery has $1,100 to invest at 9% per year compounded monthly, how long will it be before he has $1,400? If the compounding is continuous, how long will it be? (Round your answers to three decimal places.)

Solve the problem. -A venture capital firm invested $2,000,000 in a new company in 1995. In 1999, they sold their stake in the company for $10,500,000. What was the average annual rate of return on their investment?

Solve the problem. -The revenue for a dot.com company is projected to double each year for the first 5 years. If the revenue for the first year is $2 million, write a function showing the revenue R after x years. What is the revenue for the fourth year?

Solve the problem. -Tracey bought a diamond ring appraised at $1,700 at an antique store. If diamonds have appreciated in value at an annual rate of 10%, what was the value of the ring 10 years ago? (Round your answer to the nearest dollar.)

Solve the problem. -Gillian has $10,000 to invest in a mutual fund. The average annual rate of return for the past five years was 12.25%. Assuming this rate, determine how long it will take for her investment to double.

Solve the problem. Round your answer to three decimals. -How long will it take for an investment to double in value if it earns 9.25% compounded continuously?

Solve the problem. Round your answer to three decimals. -How long will it take for an investment to triple in value if it earns 8.5% compounded continuously?

Solve the problem. -The bacteria in a container quadruples every day. If there are initially 100 bacteria, write an equation that models the number of bacteria A after d days. How many bacteria will there be after 1 week?

Solve the problem. -How much money needs to be invested now to get $2000 after 4 years at 8% compounded quarterly? Express your answer to the nearest dollar.

Solve the problem. -Cindy will require $12,000 in 5 years to return to college to get an MBA degree. How much money should she ask her parents for now so that, if she invests it at 11% compounded continuously, she will have enough for School? (Round your answer to the nearest dollar.)