4500 invested at 8% continuously how long will it take to reach 9100
The correct answer and explanation is:
Let’s solve the problem step-by-step:
Problem:
You invest $4500 at an 8% annual interest rate, compounded continuously. How long will it take for the investment to grow to $9100?
Formula for continuous compounding:
A=PertA = P e^{rt}
- AA = amount after time tt
- PP = principal amount (initial investment)
- rr = annual interest rate (in decimal)
- tt = time in years
- ee = Euler’s number (~2.71828)
Given:
- P=4500P = 4500
- A=9100A = 9100
- r=8%=0.08r = 8\% = 0.08
- t=?t = ?
Step 1: Plug values into the formula
9100=4500×e0.08t9100 = 4500 \times e^{0.08t}
Step 2: Solve for tt
Divide both sides by 4500: 91004500=e0.08t\frac{9100}{4500} = e^{0.08t} 2.0222=e0.08t2.0222 = e^{0.08t}
Step 3: Take the natural logarithm (ln) of both sides
ln(2.0222)=0.08t\ln(2.0222) = 0.08t
Calculate ln(2.0222)\ln(2.0222): ln(2.0222)≈0.7031\ln(2.0222) \approx 0.7031
Step 4: Solve for tt
t=0.70310.08=8.79 yearst = \frac{0.7031}{0.08} = 8.79 \text{ years}
Final answer:
It will take approximately 8.79 years for the investment to grow from $4500 to $9100 at an 8% continuous interest rate.
Explanation:
Continuous compounding means interest is added constantly at every moment rather than at discrete intervals like yearly or monthly. This leads to faster growth compared to simple or discrete compounding.
The formula A=PertA = P e^{rt} represents this continuous growth, where ee is the base of the natural logarithm, about 2.71828, which naturally arises in continuous growth or decay processes.
To find the time tt needed to reach a certain amount AA, you rearrange the formula and use logarithms. Taking the natural logarithm isolates the variable tt, making it solvable algebraically.
In this problem, you started with 4500 and want it to grow to 9100 at 8% interest compounded continuously. After calculating, it takes about 8.79 years, which shows continuous compounding’s effect over time.
This method is widely used in finance and economics because many natural processes and investments grow in a continuous manner, making it more accurate than simple or periodic compounding formulas.