An initial amount of
$2900
is invested in an account at an interest rate of
4%
per year, compounded continuously. Find the amount in the account after six years. Round your answer to the nearest cent.
To find the amount in the account after six years, we can use the formula for continuous compound interest:
A = P * e^(r*t)
Where:
A = the final amount in the account
P = the initial amount ($2900)
e = Euler's number (approximately 2.71828)
r = interest rate per year (4% or 0.04)
t = time in years (6)
Substituting the values into the formula, we have:
A = 2900 * e^(0.04 * 6)
Calculating the exponent first:
0.04 * 6 = 0.24
Then, rewriting the formula with the calculated exponent:
A = 2900 * e^(0.24)
Using a calculator to find e^(0.24):
e^(0.24) ≈ 1.271444
Finally, calculating the final amount in the account:
A = 2900 * 1.271444 = $3687.36
Therefore, the amount in the account after six years, rounded to the nearest cent, is $3687.36.
To find the amount in the account after six years, we can use the formula for continuous compounding:
A = P * e^(rt)
Where:
A = the amount in the account after time t
P = the principal amount (initial investment)
e = the mathematical constant approximately equal to 2.71828
r = interest rate per year (in decimal form)
t = time in years
In this case, P = $2900, r = 4% = 0.04, and t = 6 years. Substituting these values into the formula, we have:
A = 2900 * e^(0.04 * 6)
Now let's calculate the value of e^(0.04 * 6):
e^(0.04 * 6) ≈ 2.22554
So, the amount in the account after six years is approximately:
A ≈ 2900 * 2.22554 ≈ $6448.84
Therefore, the amount in the account after six years, rounded to the nearest cent, is $6448.84.