Respuesta :
Answer:
It will take 28 years for Aunt Ruth to exhaust her funds.
Explanation:
This can be calculated using the formula for calculating the present value (PV) of annuity due given as follows:
PV = P * ((1 - [1 / (1 + r))^n) / r) * (1 + r) .................................. (1)
Where;
PV = Present value or amount invested = $540,000
P = Monthly withdraw = $40,000
r = interest rate = 6.5%, or 0.065
n = number of years = ?
Substitute the values into equation (1) and solve for n, we have:
540,000 = 40,000 * ((1 - (1 / (1 + 0.065))^n) / 0.065) * (1 + 0.065)
540,000 / 40,000 = ((1 - (1 / 1.065)^n) / 0.065) * 1.065
13.50 = ((1 - 0.938967136150235^n) / 0.065) * 1.065
13.50 / 1.065 = (1 - 0.938967136150235^n) / 0.065
12.6760563380282 = (1 - 0.938967136150235^n) / 0.065
12.6760563380282 * 0.065 = 1 - 0.938967136150235^n
0.823943661971833 = 1 - 0.938967136150235^n
0.938967136150235^n = 1 - 0.823943661971833
0.938967136150235^n = 0.176056338028167
Log linearizing, we have:
n log0.938967136150235 = log0.176056338028167
n = log0.176056338028167 / log0.938967136150235
n = -0.754348335711024 / -0.0273496077747565
n = 27.5816875299865
Rounding to a whole number, we have:
n = 28
Therefore, it will take 28 years for Aunt Ruth to exhaust her funds.
