Final answer:
To find how long it will take for the investment to grow to $8,000 with continuous compound interest, we can use the formula A(t) = Pe^(rt), where A(t) is the amount of money at time t, P is the principal amount, e is Euler's number, r is the interest rate, and t is the time in years. In this case, it will take approximately 8.3 years for the investment to grow to $8,000.
Step-by-step explanation:
To find how long it will take for the investment to grow to $8,000, we can use the continuous compound interest formula: A(t) = Pe^(rt), where A(t) is the amount of money at time t, P is the principal amount, e is Euler's number (approximately 2.718), r is the interest rate, and t is the time in years.
In this case, P = $4,000, A(t) = $8,000, and r = 8% (or 0.08 as a decimal). We need to solve for t. Plugging in these values, we get:
$8,000 = $4,000e^(0.08t).
Dividing both sides of the equation by $4,000, we get:
2 = e^(0.08t).
Taking the natural logarithm (ln) of both sides, we get:
ln(2) = ln(e^(0.08t)).
Using the property of logarithms, ln(e^(0.08t)) simplifies to 0.08t * ln(e), which further simplifies to 0.08t. So, we have:
ln(2) = 0.08t.
Dividing both sides of the equation by 0.08, we get:
t = ln(2) / 0.08. Using a calculator, we find that t is approximately 8.3138 years.
Rounding to the nearest tenth of a year, it will take approximately 8.3 years for the investment to grow to $8,000.