Final answer:
To find out how long you need to invest $1000 at 2.5% interest compounded continuously to reach $1100, use the continuous compounding formula A = Pe^(rt) and solve for t.
It takes approximately 3.901 years to reach the desired amount.
Step-by-step explanation:
To determine for how long we need to invest $1000 at a rate of 2.5% compounded continuously to obtain a final amount of $1100, we can use 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 (decimal), t is the time the money is invested for, and e is the base of the natural logarithm, approximately equal to 2.71828).
Plugging our values into this formula, we have 1100 = 1000e0.025t. To solve for t, we need to isolate it on one side of the equation.
This involves taking the natural logarithm of both sides of the equation.
Step by step, this would be:
- Divide both sides by 1000: 1.1 = e0.025t
- Take the natural logarithm of both sides: ln(1.1) = ln(e0.025t)
- Simplify the right side using the property of logarithms: ln(1.1) = 0.025t
- Divide by 0.025: t = ln(1.1) / 0.025
Using a calculator, we find that t ≈ 3.901 years.
This is the time needed to grow the $1000 to $1100 with continuous compounding at a 2.5% annual interest rate.