Question

Suppose that $2000 is deposits 2% compounded qauterly.

a) how much money will be in the account at the end of 7 years? (Assume withdrawals are made)

b) how long will it take for the account to grow $3000

I need this answer by 11:59 tonight

Answer 1

a) $2,299.75

b) 20.32 years

Step-by-step explanation:

Part (a)

To calculate how much money will be in an account when the interest is compounded quarterly, we can use the compound interest formula:

`\boxed{\begin{array}{l}\underline{\textsf{Compound Interest Formula}}\\\\A=P\left(1+(r)/(n)\right)^(nt)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{$A$ is the final amount.}\\\phantom{ww}\bullet\;\;\textsf{$P$ is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{$r$ is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{$n$ is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{$t$ is the time (in years).}\end{array}}`

In this case:

• P = $2,000
• r = 2% = 0.02
• n = 4 (quarterly)
• t = 7

Substitute these values into the formula and solve for A:

`A=2000\left(1+(0.02)/(4)\right)^(4 \cdot 7)`

`A=2000\left(1.005\right)^(28)`

`A=2000\left(1.14987260998...\right)`

`A=2299.74521996...`

`A=2299.75\;\sf(2\;d.p.)`

Therefore, there will be $2,299.75 in the account at the end of 7 years.

`\hrulefill`

Part (b)

To find how long it will take for the account to grow to $3,000, we can substitute the following values into the compound interest formula and solve for t:

• A = $3,000
• P = $2,000
• r = 2% = 0.02
• n = 4 (quarterly)

Therefore:

`3000=2000\left(1+(0.02)/(4)\right)^(4t)`

`3000=2000\left(1.005\right)^(4t)`

`(3000)/(2000)=\left(1.005\right)^(4t)`

`1.5=1.005^(4t)`

Take natural logs of both sides of the equation:

`\ln\left(1.5\right)=\ln\left(1.005^(4t)\right)`

Apply the power log rule:

`\ln\left(1.5\right)=4t\ln\left(1.005\right)`

Divide both sides by 4ln(1.005):

`t=(\ln\left(1.5\right))/(4\ln\left(1.005\right))`

`t=20.323896413...`

`t=20.32\;\sf years\;(2\;d.p.)`

Therefore, it will take approximately 20.32 years for the account to grow to $3,000.

Answer 2

At the end of 7 years, there will be approximately $2297.40 in the account compounded quarterly at a 2% interest rate. It will take approximately 14.21 years for the account to grow to $3000 with the same interest conditions.

Step-by-step explanation:

Calculation of Compound Interest

When solving compound interest problems, the formula A = P(1 + r/n)^(nt) is used, 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 (in decimal form).
• n is the number of times that interest is compounded per year.
• t is the time the money is invested for, in years.

For the given problem:

• P = $2000
• r = 2% or 0.02
• n = 4 (since the interest is compounded quarterly)
• t = 7 years

Thus, the equation becomes:

A = 2000(1 + 0.02/4)^(4*7)

Calculating this:

A = 2000(1 + 0.005)^(28)

A = 2000(1.005)^28

A ≈ 2000 * 1.148698

A ≈ $2297.40

So, at the end of 7 years, there will be approximately $2297.40 in the account.

Determining Time for Account Growth

To determine how long it will take for the account to grow to $3000, we rearrange the compound interest formula to solve for t:

A = 2000(1 + 0.005)^4t

Here A is $3000, and we solve for t:

3000 = 2000(1.005)^4t

1.5 = (1.005)^4t

Using logarithms, we get:

log(1.5) = 4t * log(1.005)

t ≈ log(1.5) / (4 * log(1.005))

t ≈ 14.2067

Therefore, it will take approximately 14.21 years for the account to grow to $3000.