Hey Chegg, I need help with this question. Please answer the questions that are

Question

Hey Chegg, I need help with this question. Please answer the questions that are in bold and please show work (step by step) so I can understand it more clearly.

A father is now planning a savings program to put his daughter through college. She is 13, she plans to enroll at the university in 5 years, and she should graduate in 4 years. Currently, the annual cost (for everything -food, clothing, tuition, books, transportation, and so forth) is $15,000, but these costs are expected to increase by 5% annually. The college requires that this amount be paid at the start of the year. She now has $7,500 in a college savings account that pays 6% annually. The father will make 6 equal annual deposits into her account; the first deposit today and sixth on the day she starts college. How large must each of the 6 payments be?
(Hint: Calculate the cost (inflated at 5%) for each year of college, then find the total present value of those costs, discounted at 6%, as of the day she enters college. Then find the compounded value of her initial $7,500 on that same day. The difference between the PV costs and the amount that would be in the savings account must be made up by the father's deposits, so find the 6 equal payments (starting immediately) that will compound to the required amount.)

Explanation / Answer

Amount required for the 4 years of graduate in college:-

15000 * (1.05)7 = $ 21106.51

15000 * (1.05)8 = $ 22161.83

Present value = 19144.22 + 20101.43 * P.V. factor for first year @ 6 % + 21106.51 * P.V. factor for second year @ 6 % + 22161.83 * P.V. factor for third year @ 6 %

   = 19144.22 + 20101.43 * 0.943 + 21106.51 * 0.89 + 22161.83 *0.84

   = $ 75500 (approx)

Future value of cash in hand = 7500 * (1.06)5 = 7500 * 1.3382 = $ 10036 (approx)

Thus, the net cash required = 75500 - 10036 = $ 65464

Therfore, annuity payment caculated as follows:-

Let Z denotes annuity payment.

Z * [FVIF 6%, 5] + Z * [FVIFA 6%, 5] = 65464

Z * (1.06)5  + Z * 5.6371 (Note 1) = 65464

1.3382 Z + 5.6371 Z = 65464

Z = 65464 / 6.9753

Z = $ 9385 (approx)

Conclusion: Each of six payment must be of $ 9385 (approx).

(Note 1):= Future value interest factor of annuity (FVIFA) = [(1 + r)T - 1] / r [Where r = rate of interest and T = time period in years].

   = (1 + 0.06)5 - 1 / 0.06

   = 1.3382 - 1 / 0.06

   = 0.3382 / 0.06 = 5.6371 (approx)

15000 * (1.05)7 = $ 21106.51

15000 * (1.05)8 = $ 22161.83

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