Final answer:
Using the formula for compound interest, it will take approximately 53.44 years for a $7000 deposit to grow to $19,000 at a 2% annual interest rate, compounded annually.
Step-by-step explanation:
To determine how long it will take for $7000 deposited into an account with a 2% annual interest rate, compounded annually, to grow to $19,000, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
In this case, A is $19,000, P is $7000, r is 0.02 (2% expressed as a decimal), and because it is compounded annually, n is 1. We want to solve for t. The equation rewrites to:
19000 = 7000(1 + 0.02/1)^(1*t)
We divide both sides by 7000:
2.7143 = (1.02)^t
Now, we need to solve for t. To do this, we take the natural logarithm of both sides:
ln(2.7143) = t * ln(1.02)
Finally, to solve for t, we divide the natural logarithm of 2.7143 by the natural logarithm of 1.02:
t = ln(2.7143) / ln(1.02) ≈ 53.44 years
Therefore, it will take approximately 53.44 years for the $7000 deposit to grow to $19,000 at a 2% annual interest rate compounded annually.