You leave $5,000 in an account for 20 years. The account compounds interest yearly at a rate of 10%. To the nearest dollar, how much money do you have at the end of the 20 years?
You leave $5,000 in an account for 20 years. The account compounds interest CONTINUOUSLY at a rate of 10%. To the nearest dollar, how much money do you have at the end of the 20 years?
5000*1.1^20
vs
5000*(e^.1)^20
1. 33637
2. 36,945
To find the amount of money you have at the end of 20 years when the account compounds interest yearly at a rate of 10%, you can use the formula:
A = P(1 + r/n)^(nt)
Where:
A is the final amount of money you will have after the given time period,
P is the principal amount (initial investment), which is $5,000 in this case,
r is the annual interest rate expressed as a decimal, which is 10% or 0.10,
n is the number of times interest is compounded per year, which is 1 in this case (since it compounds annually),
and t is the number of years the money is invested for, which is 20 in this case.
Plugging in the values, the equation becomes:
A = 5000(1 + 0.10/1)^(1*20)
Simplifying inside the parentheses first:
A = 5000(1 + 0.10)^20
Then, raising the result inside the parentheses to the power of 20:
A = 5000(1.10)^20
Calculating (1.10)^20:
A = 5000(6.7275)
Multiplying 5000 by 6.7275:
A ≈ 33,638
Therefore, you would have approximately $33,638 at the end of 20 years when the account compounds interest yearly at a rate of 10%.
Now, to find the amount of money you have at the end of 20 years when the account compounds interest continuously at a rate of 10%, you can use the formula for continuous compounding:
A = P * e^(rt)
Where:
A is the final amount of money you will have after the given time period,
P is the principal amount (initial investment), which is $5,000 in this case,
e is Euler's number (approximately 2.71828),
r is the annual interest rate expressed as a decimal, which is 10% or 0.10,
and t is the number of years the money is invested for, which is 20 in this case.
Plugging in the values, the equation becomes:
A = 5000 * e^(0.10 * 20)
Calculating the exponent (0.10 * 20):
A = 5000 * e^(2)
Calculating e^2:
A ≈ 5000 * 7.389
Multiplying 5000 by 7.389:
A ≈ 36,945
Therefore, you would have approximately $36,945 at the end of 20 years when the account compounds interest continuously at a rate of 10%.