A person deposits money into a retirement account, which pays 7% interest compounded continuously, at a rate of $1000 per year for 20 years. Calculate:
a. The balance of the account at the end of 20 years
b. the amount of money actually deposited into the account
c. the interest earned during the 20 years.
I think i know that for part a you use the integral to find the future value maybe, but i have no idea how to do b or c
compound interest formula is:
FV=PV(1+(r/n))^(nT)
As n approaches infinity, this formula becomes:
FV=PV*e^(rt)
for the 1st $1000 invested:
FV=1000*e^(.07*20)
for the 2nd $1000 invested:
FV=1000*e^(.07*19)
and so on for 20 years. The sum total of future values minus $20,000 deposited is the interest earned in the 20 years.
a.) $43,645.71
b.) just do 1000*20 =20,000
c.) a-b (43,645.71-20,000=23645.71)
To calculate the various values in this scenario, we can use the formula for continuously compounded interest:
A = P * e^(rt)
Where:
A = the future value or balance of the account
P = the principal amount (initial deposit)
e = Euler's number (~2.71828)
r = interest rate per time period
t = number of time periods
Let's use this formula to solve each part of the question:
a. The balance of the account at the end of 20 years:
Since the person deposits $1000 per year for 20 years, the total principal amount will be $1000 * 20 = $20,000.
Using the formula above, we can calculate the balance of the account (A) at the end of 20 years:
A = P * e^(rt)
= $20,000 * e^(0.07 * 20)
≈ $20,000 * e^(1.4)
≈ $20,000 * 4.055199966844675
≈ $81,103.99
Therefore, the balance of the account at the end of 20 years is approximately $81,103.99.
b. The amount of money actually deposited into the account:
The person deposits $1000 per year for 20 years, so the total amount deposited over this period will be $1000 * 20 = $20,000.
Therefore, the amount of money actually deposited into the account is $20,000.
c. The interest earned during the 20 years:
To calculate the interest earned, we need to find the difference between the balance of the account at the end of 20 years and the total amount deposited.
Interest earned = Balance - Deposits
= $81,103.99 - $20,000
= $61,103.99
Therefore, the interest earned during the 20 years is $61,103.99.
To solve this problem, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the balance of the account at the end of the specified time period
P = the principal amount (initial deposit)
e = the base of the natural logarithm, approximately 2.71828
r = the interest rate per year (in decimal form)
t = the number of years
a. To find the balance of the account at the end of 20 years, we need to calculate A.
Given:
r = 7% = 0.07 (decimal form)
t = 20 years
We can substitute these values into the formula:
A = P * e^(rt)
A = P * e^(0.07 * 20)
Now we need to find the value of e^(0.07 * 20). Let's calculate it:
e^(0.07 * 20) ≈ 2.71828^(0.07 * 20)
Using a calculator or a mathematical software, we find:
e^(0.07 * 20) ≈ 3.46737
Therefore, the balance of the account at the end of 20 years is:
A = P * e^(0.07 * 20)
= P * 3.46737
b. To find the amount of money actually deposited into the account, we need to calculate P. We know that the deposit is $1000 per year for 20 years:
P = $1000 * 20
P = $20,000
Therefore, the amount of money actually deposited into the account is $20,000.
c. To find the interest earned during the 20 years, we need to subtract the principal amount from the balance of the account:
Interest earned = A - P
Using the values calculated in parts a and b, we can find the interest earned.