Final answer:
To find out how long it will take for $1000 to grow to $5000 at an interest rate of 3.5% compounded continuously, use the formula for continuous compound interest and solve for time. The answer is approximately 8.36 years.
Step-by-step explanation:
To find out how long it will take for $1000 to grow to $5000 at an interest rate of 3.5% compounded continuously, we can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
- A is the final amount ($5000)
- P is the initial amount ($1000)
- r is the interest rate (0.035)
- t is the time in years (which we want to find)
Plugging in the values:
5000 = 1000 * e^(0.035t)
Simplifying the equation:
e^(0.035t) = 5
Taking the natural logarithm (ln) of both sides:
0.035t ln(e) = ln(5)
ln(e) is equal to 1:
0.035t = ln(5)
Dividing both sides by 0.035:
t = ln(5) / 0.035
Using a calculator to evaluate the expression:
t ≈ 8.36 years
Therefore, the answer is approximately 8.36 years. So the correct option is b) Approximately 8.36 years.