Final answer:
To find the interest rate as a percentage at which a $12,000 investment doubles after 7 years with continuous compounding, we can use the formula A = P * e^(rt). The interest rate in this case is approximately 9.94%.
Step-by-step explanation:
To find the interest rate as a percentage at which a $12,000 investment doubles after 7 years with continuous compounding, we can use the formula A = P * e^(rt), where A is the final amount, P is the principal, e is the base of the natural logarithm, r is the interest rate, and t is the time period. In this case, the final amount A is $24,000, the principal P is $12,000, and the time period t is 7 years. Substituting these values into the formula, we get:
$24,000 = $12,000 * e^(7r)
Dividing both sides by $12,000, we have:
2 = e^(7r)
To isolate the interest rate r, we take the natural logarithm of both sides:
ln(2) = 7r
Finally, dividing both sides by 7, we can determine the interest rate r as a decimal, and then convert it to a percentage:
r = ln(2)/7 ≈ 0.0994
Converting to a percentage, the interest rate is approximately 9.94%.