To solve the problem, we need to find out how much longer it would take for Samir's money to double compared to Penelope's money, given that Penelope invested $89,000 in an account with a continuous interest rate of 6%, while Samir invested $89,000 in an account with a monthly compounded interest rate of 6⅜%.
For Penelope's investment, we can use the formula for continuous compounding, which is A = Pe^(rt), where A is the amount of money after t years, P is the initial investment, r is the interest rate as a decimal, and e is the natural logarithm base. We know that Penelope invested $89,000 and we want to find t such that A = 2P = $178,000. Thus, we have:
$178,000 = $89,000e^(0.06t)
Dividing both sides by $89,000 and taking the natural logarithm of both sides, we get:
ln(2) = 0.06t
Solving for t, we get:
t = ln(2)/0.06 ≈ 11.55 years
For Samir's investment, we can use the formula for monthly compounded interest, which is A = P(1 + r/12)^(12t), where A, P, r are the same as before, and t is the time in years divided by 12. Similarly, we know that Samir invested $89,000 and we want to find t such that A = 2P = $178,000. Thus, we have:
$178,000 = $89,000(1 + 0.0638/12)^(12t)
Dividing both sides by $89,000 and taking the logarithm (base 1 + r/12) of both sides, we get:
log(2)/log(1 + 0.0638/12) = 12t
Solving for t, we get:
t ≈ 11.80/12 = 0.98 years
To find the difference in time it takes for Samir's money to double compared to Penelope's, we subtract the time it takes for Penelope's money to double from the time it takes for Samir's money to double:
0.98 - 11.55 ≈ -10.57
However, this answer doesn't make sense in the context of the problem, since it's negative. After reviewing our solution, we realized that we made a mistake in the calculation of t for Penelope's investment. We need to find the time it takes for Penelope's investment to double with annual compounding, not continuous compounding. The formula for this is t = (ln(2))/(ln(1 + r)), where r is the annual interest rate as a decimal.
Plugging in the numbers, we get:
t = (ln(2))/(ln(1 + 0.06)) ≈ 11.55 years
This is the same as the time we got for Samir's investment, so the difference in time it takes for their money to double is:
0.98 - 11.55 ≈ -10.57
Again, this answer doesn't make sense in the context of the problem, since it's negative. Therefore, we need to revise our solution and approach the problem differently.