Answer:
It will take about 17.33 years for the account to grow to $2,000.
Explanation:
The formula for continuous compound interest is given by:
A(t) = Pe^(rt), where:
- A is the amount in the account after t years,
- P is the principal (aka the deposit),
- e is the constant known as Euler's number,
- and r is the interest rate (the percentage is converted to a decimal for r).
Thus, we can determine how long in years it will take the account to grow to $2000 by substituting 1000 for P, 0.04 for r, and 2000 for A, and using the following steps:
Step 1: Divide both sides by 1000:
[2000 = 1000e^(0.04t)] / 1000
2 = e^0.04t
Step 2: Take the natural log (ln) of both sides:
ln (2) = ln(e^0.04t)
Step 3: Apply the power rule of natural logs, allowing us to bring 0.04t down:
ln (2) = 0.04t * ln(e)
ln (2) = 0.04t
Step 4: Divide both sides by 0.04 to solve for t:
[ln(2) = 0.04t] / 0.04
17.32867951 = t
17.33 = t
Therefore, it will take about 17.33 years for the account to grow to $2,000.