pre calc question

Question

If 4000 dollars is invested in a bank account at an interest rate of 6 per cent per year,

Find the amount in the bank after 6 years if interest is compounded annually:    

Find the amount in the bank after 6 years if interest is compounded quarterly:   

Find the amount in the bank after 6 years if interest is compounded monthly:   

Finally, find the amount in the bank after 6 years if interest is compounded continuously: 

Answer 1

Great question!

We use the compound interest formula.

A = P*(1+ r/n)^(n*t)

A = final amount

P = principal

r = interest rate

n = number of times interest applied per time period

t = time elapsed

1. A = 4000*(1+.06)^6
2. A = 4000*(1+.06/4)^(4*6)
3. A = 4000*(1+.06/12)^(12*6)

Now for the final part we need to use the compound continuous formula.

A = P*e^(r*t)

A = final amount

P = principal

e = Euler's number

r = interest rate

t = time elapsed

4.A = 4000*e^(.06*6)

Hope this helps!

Answer 2

We can use the formula for compound interest to calculate the amount in the bank after 6 years, with different compounding periods.

The formula is:

A = P(1 + r/n)^(nt)

where A is the amount, P is the principal (initial amount invested), r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time in years.

Using P = $4000, r = 6% = 0.06, and t = 6, we get:

1. Annually compounded:

n = 1

A = 4000(1 + 0.06/1)^(1*6) = $5,665.32

1. Quarterly compounded:

n = 4

A = 4000(1 + 0.06/4)^(4*6) = $5,698.74

1. Monthly compounded:

n = 12

A = 4000(1 + 0.06/12)^(12*6) = $5,725.89

1. Continuously compounded:

n = infinity

A = Pe^(rt) = 4000e^(0.06*6) = $5,759.16

Therefore, the amount in the bank after 6 years with annual, quarterly, monthly, and continuous compounding are $5,665.32, $5,698.74, $5,725.89, and $5,759.16, respectively.