Question

6. Two funds are started with equal deposits at time t = 0. Fund X is credited with simple interest at an annual rate of 3%. Fund Y is credited with force of interest of 2.9%. Let Ax (t) and Ay (t) be the amounts in funds X and Y at time t, respectively. Find the value for t at which Ax (t) – Ay (t) is maximized. Be sure to justify why the value you found is an absolute maximum.

Answer 1

the sopool 3). Initica let $x be amount inrested in given funds X and Y. Then a Here x is a fund with simple interest know, a Final value of simple interest is given by, F. V = P. V Cert) Here, Present value P.V=$x and do 31. = Also F. Va Arct) we 3 100 50, AISO) Arc Ct) = be CI+0.03+) we Know furfiese value with force or interest ris given by , e=2.7182.. F.V. where = p. v cert) an irrational constant Hese, p.v. = $x and De 2.9.1. = 0.029 F.v. = Ay Ct). Ayct) = 20.029 and so 501 ACCE Ay Ct) = 26 (1+0.035) OC 20.029+ Ax Ct) - Ay Ct) = 20 CI+0.03t - 20.029+) ,

FCU- - How, say Arct) - Ayct) Then, F°C+) = 2 (1+0.03t - €0.029+) F'C+ = X co+ 0.03 - 0.029 e + (p=0, 423) - pfct) dfc+) for cury =) 0.029+ dt constant c c) ) F'Ct) = a Co.o3 -0.029 e 0.029t How, for stationary point, F'(t) = 0 ac e ie 2c (0.03 - 0.029 €0.orgt ) =0 0.03 = 0.029 e 0.02gt (xto as initial amount Oc0297 is not o е e 0.029 1.0348483 = 20.029+ en (1.036483) = 0.029 + ea=b =) en b= = . =) =) 0.03 Q 긴 닌 T ㄷ t = 0.03390 I 0.029 € 0.03390 1.169019 o.org tĚ 1.169 is coitical point. ترانه ۴۱ A(Z) = ( ( ac dt 0.029 C0.029) €0.0294 eft afct); fer comes to =o defect) ct constant c

= ) F"(t) = -0.000841 0.029t e = F"(1.169) - 0000841 €0.029(1.169) 0.00086999 < 0 -0.000841 C1.034982) F1 C1:169) second derivative test t = 1.169 is point of absolute musiimum. masimum is required value of t =) . By :. t=1-169