Final answer:
To determine the time it takes for an investment to grow to a certain amount with continuous compounding, use the formula A = Pe^rt. For the $1000 growing to $1800 at a rate of 4.9%, it takes approximately 12 years.
Step-by-step explanation:
The question you've asked is about determing the number of years it will take for an investment to grow to a certain amount when the interest is compounded continuously. We can solve this problem using the formula for continuous compounding, which is A = Pert, where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate (expressed as a decimal), and t is the time in years.
In your case, you want to find out when $1000 (the principal P) becomes $1800 (the accumulated amount A) with a continuous interest rate of 4.9% or 0.049 (the rate r). Reworking the formula to solve for t, we get t = (ln(A/P)) / r. Plugging in the values, we get t = (ln(1800/1000)) / 0.049. Now we just need to calculate it.
Calculate t step by step:
- Divide 1800 by 1000 to get 1.8
- Take the natural logarithm of 1.8 to get approximately 0.5878
- Divide 0.5878 by 0.049 to get the number of years, which is approximately 12 years
It will take approximately 12 years for your $1000 investment to grow to $1800 with continuous compounding at a rate of 4.9%.