Meg's pension plan is an annuity with a guaranteed return of 6% per year (compounded quarterly). She would like to retire with a pension of $50,000 per quarter for 15 years. If she works 31 years before retiring, how much money must she and her employer deposit each quarter? HINT [See Example 5.] (Round your answer to the nearest cent.)

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To find out how much money Meg and her employer must deposit each quarter, we can use the formula for the periodic payment on an annuity.

The formula is:

P = (PV * r) / (1 - (1 + r)^(-n))

Where:
P = Periodic Payment
PV = Present Value (the amount of money needed to fund the desired pension)
r = Interest rate per period
n = Number of periods

In this case, Meg wants a pension of $50,000 per quarter for 15 years, which is a total of 15 * 4 = 60 quarters.

We need to find the Present Value (PV), which is the amount of money needed to fund the desired pension. To do this, we'll rearrange the formula:

PV = (P * (1 - (1 + r)^(-n))) / r

Now we can substitute the values into the formula:

PV = ($50,000 * (1 - (1 + 0.06/4)^(-60))) / (0.06/4)

Simplifying the equation inside the brackets:

PV = ($50,000 * (1 - (1.015)^(-60))) / (0.015)

Calculating the value inside the brackets:

PV = ($50,000 * (1 - 0.232545)) / (0.015)

PV = ($50,000 * 0.767455) / (0.015)

PV = $3,837,275.17

Now, since Meg works for 31 years, the total number of quarters Meg and her employer will deposit money for is 31 * 4 = 124 quarters.

Finally, we can find the amount that Meg and her employer must deposit each quarter by rearranging the formula:

P = (PV * r) / (1 - (1 + r)^(-n))

P = ($3,837,275.17 * 0.06/4) / (1 - (1 + 0.06/4)^(-124))

Calculating the value inside the brackets:

P = ($3,837,275.17 * 0.015) / (1 - (1.015)^(-124))

Simplifying:

P = $57,558.628 / (1 - 0.120181)

P = $57,558.628 / 0.879819

P = $65,472.09

Therefore, Meg and her employer must deposit approximately $65,472.09 each quarter to accumulate enough money to provide a pension of $50,000 per quarter for 15 years.