Question

a salt concentration of 1+1 sin t) g/L flows into the tank at a rate of 3 L/min, and the mixture flows out at the same rate. 5. Continuously Compounded Interest. The amount of money A(t) in an interest bearing account in which the principal is compounded continuously at a rate r per year and in which money is continuously added, or subtracted, at a rate of k dollars per year satisfies the differential equation dA (15) (a) What does k<0 or k>0 correspond to? (b) Show that the solution to equation (15) with initial condition A(0)=Ao is (16)

Answer 1

(a) k < 0 implies money is continuously subtracted at a rate k dollar per year k > 0 implies money is added continuously at a rate k dollar per year (b) The given diffn equation is dA/dt = rA + k dA/rA + k = dt integrating this between when a = A_0 then t = 0 to A = A the t = t integral^A_A_0 dA/rA + k = integral^t_0 dt 1/r [log (rA + k)]^A_A_0 = t -0 log(rA + k/rA_0 + k) = rt rA + k/rA_0 + k = e^rt rA + k = rA_0e^rt + ke^rt rA = e^rt (r A_0 + k) - k A(t) = e^rt (A_0 + k/r) - k/r (proved) (d) Given A(t + h) = A(t) + r A(t) h + k h A(t + h) - A(t)/h = r A(t) + k taking lim h rightarrow 0 we have lim_h rightarrow 0 A(t + h) - A(t)/h = rA(t) \r\n