Earning interest- You deposit $1000 in an account that pays 6% annual interest compounded continuously. Find the balance at the end of 2 years.
1,000 * e^(.06 *2) = 1318.26
because
dx/dt = .06 x
dx/x = .06 dt
ln x = .06 t
x = e^.06 t + c
or
x = Xo e^.06 t
where Xo is initial amount because e^0 = 1
To find the balance at the end of 2 years, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the balance at the end of the specified time period
P = the principal amount (initial deposit)
e = the mathematical constant approximately equal to 2.71828
r = the annual interest rate (in decimal form)
t = the time period in years
In this case, the principal amount is $1000, the annual interest rate is 6% (0.06 in decimal form), and the time period is 2 years. Plugging in these values into the formula:
A = 1000 * e^(0.06*2)
Using a calculator, we can evaluate e^(0.06*2) ≈ 1.123716. So:
A = 1000 * 1.123716
A ≈ $1123.72
Therefore, the balance at the end of 2 years would be approximately $1123.72.
To find the balance at the end of 2 years, we can use the formula for compound interest with continuous compounding:
A = P * e^(rt)
Where:
A = the final amount
P = the initial amount (principal)
e = Euler's number, approximately 2.71828
r = the interest rate
t = the time in years
In this case, P = $1000, r = 6% = 0.06 (in decimal form), and t = 2.
A = $1000 * e^(0.06 * 2)
Now, let's calculate it step by step:
Step 1: Calculate the exponent term (0.06 * 2):
0.06 * 2 = 0.12
Step 2: Calculate e^(0.12) using a calculator:
e^(0.12) ≈ 1.1275
Step 3: Multiply the result by the initial amount:
A = $1000 * 1.1275
Step 4: Calculate the final amount:
A ≈ $1127.50
Therefore, the balance at the end of 2 years, with continuous compounding and a 6% annual interest rate, would be approximately $1127.50.