Final answer:
Mary will accumulate approximately $35,947.35 by her 58th birthday through quarterly contributions of $200 at a 5% interest rate compounded semi-annually.
Step-by-step explanation:
To calculate the amount Mary will accumulate by her 58th birthday, we need to determine the total amount she will contribute and the interest she will earn over the years. Mary plans to contribute $200 at the end of every quarter year, so she will make a total of $800 in contributions each year. Since the interest is compounded semi-annually, we need to divide the annual interest rate by 2. With a 5% annual interest rate, the semi-annual interest rate is 2.5%.
To determine the accumulated amount, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A is the accumulated amount
P is the principal (initial contribution)
r is the annual interest rate
n is the number of times interest is compounded per year
t is the number of years
In this case, the principal is $200, the annual interest rate is 5%, the semi-annual interest rate is 2.5%, the number of times interest is compounded per year is 2, and the number of years is 58 - 33 = 25.
Plugging these values into the formula, we get:
A = 200 * (1 + 0.025/2)^(2*25)
Calculating this using a calculator, we find that Mary will accumulate approximately $35,947.35 by her 58th birthday.