Question

1. Professor Rapatski and her husband were (pleasantly) surprised by their third addition Theodore. They had already been saving for Joey and Andy's college fund, but needed to start Theo's. They decided the best option would be to sell Dr. Rapatski's beloved 1972 Volkswagen Beetle for $15,000. They will start a NJ Best 529 College Savings plan with the $15,000 and deposit an additional $200 a month ($2400 a year). This College Savings Plan gains 8% interest a year. Assume interest and deposits are continuous. a) Write a differential equation to represent the amount in the college fund at timet with an initial condition. Solve and determine how much money there will be in the savings account when Theo is ready to start college (in 18 years assuming they started saving from Theo's birth). b) Professor Rapatski just didn't have the heart to sell her car. Instead she opened the account with $3300 from her personal savings. To make up the difference they decided that they will contribute an extra $150 a month until the $11,700 is paid back. So for 6.5 years they will contribute $350 each month. After which they will return to depositing just $200 a month. How much money will the college savings account have after 18 years? Why are the answers not the same? By keeping her car, how much less is the college fund? C)

Answer 1

2 1. S 1 1 ) ir Alt) be the amount of money in bank after oto years. a.) 60, A(0) = $15,000 ie starting amount guere t- yeau, we have $15,000 plus $24,00 deposit So, ACH) = (15000 +8400) + 8 (15000+2400) = (15000 + 2400) (-+ o) 15000 (it ede)* + $400 (hte edhe aftere 2-year A(2) = (A(1) +Q400) + 8 (ALL)+2400) 1 1 1 1 1 1 0 100 0 0 1 0 | | | | +0400 | | • (ALL) +8400) (+ = [(15000 + Suco) (19&) +2400](") = 15000 (1+ )+ Q400 (1 + 12 +4400 (14 eur) (15000+&+00)(4+ * +2400 (it ole 3-year A(3) - (Al2)+8400) + % ( A12)+8400) = (Aca) + 4400) (1+0) - A(2) (1+PB) + $400 (it Poo). • (15000 +2400)(1+ 8.)* + 2400 (10 6. J° +2400(1) | | afrer | ' ' 1 1 1 1 0

0 0 tl.509991$ ñ prosh 81 morto 001 Ti-gilet corzę+ 311 801) ooht) = 1814 AIO)- 15000 Qol 801/us [0036+ ] ( ) wr ,( ) of ) oohp2 + (501) W 7 (801) cont! - HP Cos [va(t) Joonce + g(**) cont] 1 Siccome + s (SM)cant! 222222 OL 0048 + 4 002 Joontl Ale! - 17u00(in yle + dvo(bare) + (ra) * 17400 10 A(t) = сам. Sо, уку е (no comp+ (50Conp +plaset cong + 1 ( ) conti - () U Promognuno go

- b) to to A101 = $ 3300 ALL= (3300 +4200) + ?(3300 +4200) = (3300 +4200 ) (108 = (7500 |1102) to 2 A(2) = (1500 ) (1.08) + 4200 + (15 (1-08)+22 loo (+fo) (150011:08) + 4200 = 7500 (1608)2 + 4200(1.08) 7500 (108)3 + 420011.08)2 +4200 (1-08) A(3) - 0,08 A(6) = 750011.08)6 + 4200 (1.08)+ (1.08)2 + = 1500 (1.08)6 +4200 x 1108 (11:08]5-1) = 7500 (1.08)6 + 56700 (( 6815-1) ñ 38 512.45 A(7) = A(6) + (6x350 + 6x 200) + 8 (116)+ 6x350 + 6x200) 1 1 45157.45 0 0 0 0 0

After 7- years, it would by $9400 per year again so, AlB) = (45157+&400) (108) 9 2 2 ) cong + ( 801) +sish ( les cond + eläi) cong + z 1 Sai) tsish 2 - 1018 2 2 1 0 -- 4801) one to let tsish = (2134 0 300 0 - 45157 (1.0819 + de 800 (1,08) (108) 9-1 0 0 008 0 [1-6680-1)] cohée + 6/80-1) tsish - +E92018 0 0 0 0 (C) 0 Difference = $ 43963. Interest rate on intial deposit is more cumulative in case of part (a) : A (= $15000 > $ 3300 0 0 1 1 1 1 1 1 1