Final answer:
To find out how long it will take for $4,000 to grow to $8,000 with 8% interest compounded continuously, we use the formula A(T) = Pert. Solving the equation 8,000 = 4,000e0.08T gives T ≈ 8.66, which means it will take approximately 8.7 years.
Step-by-step explanation:
To determine how long it will take for an investment of $4,000 at 8% interest compounded continuously to grow to $8,000, we can use the continuous compounding formula A(T) = Pert. Here, A(T) represents the amount of money accumulated after time T, P is the principal amount, e is the base of the natural logarithm, r is the rate of interest, and T is the time in years.
According to the given problem:
- Final amount A(T) = $8,000
- Principal P = $4,000
- Rate r = 8% or 0.08
We can set up the equation 8,000 = 4,000e0.08T and solve for T.
First, divide both sides by 4,000 to get e0.08T = 2. Next, take the natural logarithm of both sides to get 0.08T = ln(2). Finally, divide by 0.08 to solve for T.
The calculation is as follows:
- e0.08T = 2
- 0.08T = ln(2)
- T = ln(2) / 0.08
- T ≈ 8.66
Therefore, it will take approximately 8.7 years for the investment to grow to $8,000.