Question 1

Assume the cost of a gallon of milk is $2.60. With continuous compounding, find the time it would take the cost to be 4 times as much (to the nearest tenth of a year), at an annual inflation rate of 6%.
A) 23.1 years
B) 35.9 years
C) 0.1 year
D) 8.5 years

Accepted Answer

The formula for continuous compounding is $A = Pe^{rt}$, where $A$ is the amount after time $t$, $P$ is the principal amount, $r$ is the rate, and $t$ is the time in years. To find the time it takes for the cost to quadruple, set $A = 4P$ (since 4 times as much as the initial amount), $P = \$2.60$ (which will cancel out), and $r = 0.06$ (6% annual inflation rate). Solve for $t$: $4 = e^{0.06t}$. Taking the natural logarithm of both sides gives $ln(4) = 0.06t$, so $t = ln(4)/0.06 \approx 23.1$ years.

Question 2

Sumi Kato's savings account has a balance of $3950. After 13 years, what will the amount of interest be at 5.5% compounded annually?
A) $5135.00
B) $3961.81
C) $3972.81
D) $3978.81

Accepted Answer

The answer of Sumi Kato's savings account has a balance...

Question 3

June made an initial deposit of $4700 in an account for her son. Assuming an interest rate of 8% compounded quarterly, how much will the account be worth in 12 years?
A) $11,920.81
B) $12,047.53
C) $12,159.23
D) $7710.85

Accepted Answer

The answer of June made an initial deposit of $4700...

Question 4

Provide an appropriate response.
-Using A $$\mathrm { A } = \mathrm { P } ( 1 + \mathrm { rt } )$$ rt), the future value formula for a simple interest investment, derive the formula for r, the rate
of simple interest.

Accepted Answer

The answer of Provide an appropriate response.
-Using A \[\mathrm {...

Question 5

How long would it take $5000 to grow to $25,000 at 9% compounded continuously? Round your answer to the nearest tenth of a year.
A) 18.4 years
B) 17.9 years
C) 19.4 years
D) 18.1 years

Accepted Answer

The answer of How long would it take $5000 to...

Question 6

Find the amount of money in an account after 5 years if $2800 is deposited at 7% annual interest compounded monthly.
A) $3969.35
B) $3961.38
C) $3949.68
D) $3927.14

Accepted Answer

The amount of money in an account after a certain period with compound interest is calculated using the formula A = P(1 + 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 this problem, P = $2800, r = 7% or 0.07, n = 12 (since the interest is compounded monthly), and t = 5 years. Plugging these values into the formula gives A = 2800(1 + 0.07/12)^(12*5) = $3969.35.

Question 7

Provide an appropriate response.
-Using A $\mathrm { A } = \mathrm { P } ( 1 + r \mathrm { t } )$ ), the future value formula for a simple interest investment, derive the formula for t, the time
(in years).

Accepted Answer

The answer of Provide an appropriate response.
-Using A \(\mathrm {...

Question 8

Barry Newman's savings account has a balance of $1541. After 17 years, what will the amount of interest be at 6% compounded annually?
A) $2608.56
B) $2599.56
C) $924.60
D) $2613.56

Accepted Answer

The amount of interest can be calculated using the formula for compound interest: A = P(1 + r/n)^(nt) - P, where A is the amount of interest, P is the principal amount ($1541), r is the annual interest rate (6% or 0.06), n is the number of times the interest is compounded per year (1 for annually), and t is the time the money is invested for (17 years). Plugging in the values: A = 1541(1 + 0.06/1)^(1*17) - 1541 = $2608.56.

Question 9

Use your graphing calculator to find how long it will take for $6600 invested at 5.85% per year compounded daily to triple in value. Find the answer to the nearest year.
A) 25,219 years
B) 19 years
C) 0 years
D) 2,502,126 years

Accepted Answer

To find how long it will take for an investment to triple in value with daily compounding interest, we use the formula $A = P(1 + \frac{r}{n})^{nt}$, 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. To triple the investment, $A = 3P$, so we have $3P = P(1 + \frac{0.0585}{365})^{365t}$. Solving for $t$ with a graphing calculator or through logarithmic manipulation, we find that it takes approximately 19 years for the investment to triple.

Question 10

Felipe Rivera's savings account has a balance of $2238. After 3 years what will the amount of interest be at 6% compounded quarterly?
A) $428.79
B) $437.79
C) $67.14
D) $442.79

Accepted Answer

The formula for compound interest is A = P(1 + r/n)^(nt). Here, P = $2238, r = 0.06, n = 4, t = 3. The interest is A - P = $437.79.

Question 11

$2500 is invested at 6% compounded quarterly. In how many years will the account have grown to $6000? Round your answer to the nearest tenth of a year.
A) 1.2 years
B) 18.5 years
C) 15.0 years
D) 14.7 years

Accepted Answer

The formula for compound interest 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. Here, $A = 6000$, $P = 2500$, $r = 0.06$, and $n = 4$ (since the interest is compounded quarterly). We solve for $t$:$$6000 = 2500(1 + \frac{0.06}{4})^{4t}$$$$2.4 = (1 + 0.015)^{4t}$$$$2.4 = (1.015)^{4t}$$Taking the natural logarithm of both sides:$$\ln(2.4) = 4t \ln(1.015)$$$$t = \frac{\ln(2.4)}{4\ln(1.015)}$$$$t \approx 14.7$$ years.

Question 12

Provide an appropriate response.
-If interest is compounded monthly at 5% per year for 14 years, explain how to find the number of compounding
periods and the interest rate per compounding period.

Accepted Answer

The answer of Provide an appropriate response.
-If interest is compounded...

Question 13

Find the amount of money in an account after 10 years if $2100 is deposited at 4% annual interest compounded annually.
A) $3126.61
B) $3130.75
C) $3120.49
D) $3108.51

Accepted Answer

The answer of Find the amount of money in an...

Question 14

An initial investment of $660 is appreciated for 11 years in an account that earns 1% interest, compounded continuously. Find the amount of money in the account at the end of the period.
A) $39,516,933.46
B) $736.74
C) $722.81
D) $1650.62

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 deposited or borrowed for, in years. Here, $P = $660$, $r = 0.01$, and $t = 11$. Plugging these values into the formula gives $A = 660e^{0.01 \times 11}$, which calculates to approximately $736.74.

Question 15

You have money in an account at 6% interest, compounded weekly. To the nearest year, how long will it take for your money to double?
A) 9 years
B) 16 years
C) 7 years
D) 12 years

Accepted Answer

The Rule of 72 is a simple way to estimate the number of years required to double the investment at a given annual rate of return. By dividing 72 by the annual rate of interest, you get an approximate number of years. For a 6% interest rate, 72 / 6 = 12 years. Although the interest is compounded weekly, the Rule of 72 gives a close approximation for annual compounding and is still useful for a rough estimate in this case.

Question 16

You have money in an account at 5% interest, compounded weekly. To the nearest year, how long will it take for your money to triple?
A) 31 years
B) 18 years
C) 22 years
D) 13 years

Accepted Answer

To find the time it takes for money to triple with compound interest, we use the formula $A = P(1 + \frac{r}{n})^{nt}$, 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. We want to find $t$ when $A = 3P$, $r = 0.05$, and $n = 52$ (since the interest is compounded weekly). Rearranging the formula to solve for $t$, we get $t = \frac{\log(A/P)}{n \cdot \log(1 + r/n)}$. Plugging in the values, we get $t \approx 22$ years.

Question 17

Southwest Dry Cleaners believes that it will need new equipment in 9 years. The equipment will cost $26,000. What lump sum should be invested today at 12% compounded semiannually, to yield $26,000?
A) $9108.94
B) $15,440.52
C) $19,593.79
D) $15,212.33

Accepted Answer

To find the present value (the lump sum to be invested today) that will grow to $26,000 in 9 years at a 12% annual interest rate compounded semiannually, we use the formula for the present value of a single future amount: $PV = \frac{FV}{(1 + r/n)^{nt}}$, where PV is the present value, FV is the future value ($26,000), r is the annual interest rate (12% or 0.12), n is the number of compounding periods per year (2 for semiannual), and t is the time in years (9). Plugging in the values, we get $PV = \frac{26,000}{(1 + 0.12/2)^{2*9}}$, which calculates to approximately $9108.94.

Question 18

What will be the amount in an account with initial principal $9000 if interest is compounded continuously at an annual rate of 7.25% for 7 years?
A) $14,950.20
B) $5066.20
C) $9676.74
D) $9000.00

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 (initial investment), $r$ is the annual interest rate (decimal), $t$ is the time the money is invested for in years, and $e$ is the base of the natural logarithm (approximately equal to 2.71828). Plugging in the given values: $A = 9000e^{0.0725*7}$, we find that the amount is closest to $14,950.20.

Question 19

Anwar invested $2500 at 4% compounded semiannually. In how many years will Anwar's investment have tripled? Round your answer to the nearest tenth of a year.
A) 3.5 years
B) 4.2 years
C) 27.5 years
D) 27.7 years

Accepted Answer

To find the time it takes for the investment to triple, use the formula for compound interest: A = P(1 + r/n)^(nt). Here, A = 3P, P = $2500, r = 0.04, n = 2. Solving for t gives approximately 27.7 years.

Question 20

Barbara knows that she will need to buy a new car in 2 years. The car will cost $15,000 by then. How much should she invest now at 8%, compounded quarterly, so that she will have enough to buy a new car?
A) $12,802.36
B) $13,888.89
C) $13,868.34
D) $11,907.48

Accepted Answer

To find the present value, use the formula for compound interest: $ P = \frac{F}{(1 + \frac{r}{n})^{nt}} $. Here, $ F = 15000 $, $ r = 0.08 $, $ n = 4 $, $ t = 2 $. Plugging in these values gives $ P = \frac{15000}{(1 + \frac{0.08}{4})^{4 \times 2}} \approx 12802.36 $.

Quiz 5: Exponential and Logarithmic Functions

Assume the cost of a gallon of milk is $2.60. With continuous compounding, find the time it would take the cost to be 4 times as much (to the nearest tenth of a year) , at an annual inflation rate of 6%.

Sumi Kato's savings account has a balance of $3950. After 13 years, what will the amount of interest be at 5.5% compounded annually?

June made an initial deposit of $4700 in an account for her son. Assuming an interest rate of 8% compounded quarterly, how much will the account be worth in 12 years?

Provide an appropriate response. -Using A $\mathrm { A } = \mathrm { P } ( 1 + \mathrm { rt } )$ rt), the future value formula for a simple interest investment, derive the formula for r, the rate of simple interest.

How long would it take $5000 to grow to $25,000 at 9% compounded continuously? Round your answer to the nearest tenth of a year.

Find the amount of money in an account after 5 years if $2800 is deposited at 7% annual interest compounded monthly.

Provide an appropriate response. -Using A $\mathrm { A } = \mathrm { P } ( 1 + r \mathrm { t } )$ ), the future value formula for a simple interest investment, derive the formula for t, the time (in years).

Barry Newman's savings account has a balance of $1541. After 17 years, what will the amount of interest be at 6% compounded annually?

Use your graphing calculator to find how long it will take for $6600 invested at 5.85% per year compounded daily to triple in value. Find the answer to the nearest year.

Felipe Rivera's savings account has a balance of $2238. After 3 years what will the amount of interest be at 6% compounded quarterly?

$2500 is invested at 6% compounded quarterly. In how many years will the account have grown to $6000? Round your answer to the nearest tenth of a year.

Provide an appropriate response. -If interest is compounded monthly at 5% per year for 14 years, explain how to find the number of compounding periods and the interest rate per compounding period.

Find the amount of money in an account after 10 years if $2100 is deposited at 4% annual interest compounded annually.

An initial investment of $660 is appreciated for 11 years in an account that earns 1% interest, compounded continuously. Find the amount of money in the account at the end of the period.

You have money in an account at 6% interest, compounded weekly. To the nearest year, how long will it take for your money to double?

You have money in an account at 5% interest, compounded weekly. To the nearest year, how long will it take for your money to triple?

Southwest Dry Cleaners believes that it will need new equipment in 9 years. The equipment will cost $26,000. What lump sum should be invested today at 12% compounded semiannually, to yield $26,000?

What will be the amount in an account with initial principal $9000 if interest is compounded continuously at an annual rate of 7.25% for 7 years?

Anwar invested $2500 at 4% compounded semiannually. In how many years will Anwar's investment have tripled? Round your answer to the nearest tenth of a year.

Barbara knows that she will need to buy a new car in 2 years. The car will cost $15,000 by then. How much should she invest now at 8%, compounded quarterly, so that she will have enough to buy a new car?