Suppose that $4000 is invested at an interest rate of 5.5% per year, compounded continuously. What is the balance after 8 years?
FV=4000e^8x.055= 4000e^=6210.83
What amount will be in an account after 1.5 years if $4,000 is invested at 5% compounded semiannually?
To find the balance after 8 years with continuous compounding, we can use the formula:
\[A = P \cdot e^{rt}\]
Where:
A is the final amount
P is the principal amount (initial investment)
r is the annual interest rate (as a decimal)
t is the time in years
e is Euler's number (approximately 2.71828)
In this case:
P = $4000
r = 5.5% = 0.055 (as a decimal)
t = 8 years
Plugging these values into the formula, we get:
\[A = 4000 \cdot e^{0.055 \cdot 8}\]
To find the balance after 8 years with continuous compounding, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the final balance
P = the initial principal (or investment)
e = Euler's number (approximately 2.71828)
r = the interest rate (expressed as a decimal)
t = the time period (in years)
In this case, the initial principal is $4000, the interest rate is 5.5% per year (or 0.055 as a decimal), and the time period is 8 years.
Plugging in these values into the formula, we have:
A = 4000 * e^(0.055 * 8)
Now, we need to evaluate the exponential term. You can use a calculator with an exponential function (often denoted as "e^x" or "exp(x)") to compute this value.
Let's calculate it step by step:
1. First, calculate 0.055 * 8:
0.055 * 8 = 0.44
2. Next, calculate e^(0.44) using a calculator. The value of e is approximately 2.71828.
e^(0.44) ≈ 2.71828^(0.44) ≈ 1.55270435038
Now, multiply the result by the initial principal to find the final balance:
A ≈ 4000 * 1.55270435038
Evaluating this expression:
A ≈ 6210.81740152
Therefore, the balance after 8 years with continuous compounding would be approximately $6210.82.