To determine how long it will take Alonzo to retire, we can use the future value of an annuity formula:
A = P * ((1 + r/n)^(nt) - 1) / (r/n)
Where:
A = accumulated value (target amount)
P = payment per period
r = interest rate
n = number of compounding periods per year
t = number of years
In this case, Alonzo plans to accumulate $250,000, with quarterly payments of $5,000. The interest rate is 5.4%, compounded quarterly.
Let's plug in the values into the formula:
250,000 = 5,000 * ((1 + 0.054/4)^(4t) - 1) / (0.054/4)
Simplifying further:
(1 + 0.054/4)^(4t) - 1 = 250,000 * (0.054/4) / 5,000
Now, we can solve for t using logarithms.
(1 + 0.054/4)^(4t) = (250,000 * (0.054/4) / 5,000) + 1
Taking the natural logarithm on both sides:
ln((1 + 0.054/4)^(4t)) = ln((250,000 * (0.054/4) / 5,000) + 1)
Now, we can bring the exponent down:
(4t) * ln(1 + 0.054/4) = ln((250,000 * (0.054/4) / 5,000) + 1)
Finally, solving for t:
t = ln((250,000 * (0.054/4) / 5,000) + 1) / (4 * ln(1 + 0.054/4))
Using a calculator, we can find the value of t to the nearest year.