Determine the amount of money that will be accumulated in an account that pays compound interest, given the initial principal of $ 29,400 invested at 2.77% annual interest for 7 years compounded
(a) daily (n365);
(b) continuously.
We can use the compound interest formula to calculate the amount of money accumulated in the account.
(a) Daily compounding:
The formula is given by A = P(1 + r/n)^(nt), where:
A = the amount of money accumulated
P = the initial principal
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
Using the given values:
P = $29,400
r = 2.77% = 0.0277
n = 365
t = 7
Plugging the values into the formula, we get:
A = $29,400 * (1 + 0.0277/365)^(365*7)
Calculating this expression gives us:
A ≈ $33,929.57
Therefore, the amount of money that will be accumulated in the account with daily compounding is approximately $33,929.57.
(b) Continuous compounding:
The formula for continuous compounding is given by A = P * e^(rt), where:
A = the amount of money accumulated
P = the initial principal
r = annual interest rate (as a decimal)
t = number of years
e = Euler's number (approximately 2.71828)
Using the given values:
P = $29,400
r = 2.77% = 0.0277
t = 7
Plugging the values into the formula, we get:
A = $29,400 * e^(0.0277*7)
Calculating this expression gives us:
A ≈ $34,023.12
Therefore, the amount of money that will be accumulated in the account with continuous compounding is approximately $34,023.12.
To determine the amount of money accumulated in an account that pays compound interest, we need to use the compound interest formula.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal (initial amount)
r = the annual interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years
For part (a) - compounded daily:
In this case, n = 365 since interest is compounded daily.
P = $29,400
r = 2.77% = 0.0277 (in decimal form)
t = 7 years
Plugging in the values into the formula:
A = $29,400 * (1 + 0.0277/365)^(365*7)
To calculate this, we need to break it down into smaller steps:
1. Divide the annual interest rate by the number of compounding periods per year:
0.0277 / 365 = 0.0000758904
2. Add 1 to the result:
1 + 0.0000758904 = 1.0000758904
3. Raise this result to the power of the total number of compounding periods:
(1.0000758904)^(365*7) ≈ 1.21377918
4. Multiply the result by the principal amount:
$29,400 * 1.21377918 ≈ $35,666.21
Therefore, the amount accumulated in the account compounded daily will be approximately $35,666.21.
For part (b) - compounded continuously:
In this case, n approaches infinity, as interest is compounded continuously.
P = $29,400
r = 2.77% = 0.0277 (in decimal form)
t = 7 years
The formula for continuously compounded interest is:
A = P * e^(rt)
Where e is Euler's number, an irrational constant approximately equal to 2.71828.
Plugging in the values into the formula:
A = $29,400 * e^(0.0277 * 7)
Calculating this:
e^(0.0277 * 7) ≈ 1.21199813
Multiplying this by the principal amount:
$29,400 * 1.21199813 ≈ $35,667.75
Therefore, the amount accumulated in the account compounded continuously will be approximately $35,667.75.
Please note that these calculations are approximate due to rounding.
To determine the amount of money accumulated in the account, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (accumulated money)
P = the initial principal ($29,400)
r = the annual interest rate (2.77% or 0.0277)
n = the number of times interest is compounded per year
a) Daily compounding (n = 365):
In this case, interest is compounded 365 times per year. Substituting the values into the formula:
A = 29,400(1 + 0.0277/365)^(365*7)
Simplifying the formula:
A ≈ 29,400(1 + 0.000076)^2555
A ≈ 29,400(1.000076)^2555
Calculating:
A ≈ 29,400 * 1.207418
A ≈ $35,526.89
So, the amount of money accumulated in the account with daily compounding over 7 years would be approximately $35,526.89.
b) Continuous compounding:
In this case, the compound interest formula changes slightly to:
A = Pe^(rt)
Substituting the values into the formula:
A = 29,400 * e^(0.0277*7)
Calculating:
A ≈ 29,400 * e^(0.1939)
A ≈ 29,400 * 1.21331
A ≈ $35,703.91
So, the amount of money accumulated in the account with continuous compounding over 7 years would be approximately $35,703.91.