Ch7 Differential Equations APQ Solutions (RVHS)

2022 JC1 H2 Mathematics 9788 RVHS Mathematics Department Ch 7 Differential Equations (Solutions to APQ) 1 CHAPTER 7 DIFFERENTIAL EQUATIONS Tutorial Solutions Additional Practice Questions 1. HCI/2003/II/2 Liquid is poured into a container at a constant rate of 30 cm3 s1 and leaks out at a rate which is proportional to the volume of liquid in the container. At time t seconds, the volume of the water in the container is V cm3. When the volume reaches 450 cm 3, it decreases at a rate of 30 cm3 s1. Show that d15 2 450.d V Vt   [2] It is given that V = 1000 cm3 when t = 0. (i) Show that tV Ae B , where , A and B are constants to be determined. [4] (ii) Sketch the graph of V against t. [1] 1 (i) in outdV dVdV dt dt dt 30dV kVdt  At V = 450, 30dV dt  : –30 = 30 – k(450) => k = 2/15 230 15 dV Vdt   => 15 450 2dV Vdt   (shown) 15 2 450 dV dtV   15 ln | 2 450 |2 V t C    When t =0, V = 1000: 15 ln15502C => 15 15ln | 2 450 | ln155022 Vt    2ln | 2 450 | ln155015 tV     2 15 2 450 1550 t Ve    2 15 775 225 t Ve     V 1000 225 0 t V = 775e–(2/15)t +225 (ii)

2022 JC1 H2 Mathematics 9788 RVHS Mathematics Department Ch 7 Differential Equations (Solutions to APQ) 2 2. RJC/2003/I/16(b) At the instant when the radius of a sphere is r cm, its volume, V cm3, is increasing at the rate of ( – r 3) cm3min–1, where  is a constant. Express dV dt in terms of dr dt and show that 4r 2 dr dt =  – r 3. [3] If the initial volume of the sphere is zero, find t when the volume is  6 cm3. [5] [Volume of a sphere = 4 3  r 3 ] 2 3()dV rdt  -----(1) 34 3Vr  => 24dV rdr  24dV dV dr dr rdt dr dt dt     Hence, from (1): 32( ) 4 drrr dt     => 234 drrr dt  (shown) 2 3 4r dr dtr  2 3 43 3 r dr t Cr     34 ln3 r t C    When t = 0, r = 0: 4 ln3 C When 6V  , 34 36 r   => 3 8r  At this instance, 44ln ln3 8 3 t    => 48ln 0.17837t   min

2022 JC1 H2 Mathematics 9788 RVHS Mathematics Department Ch 7 Differential Equations (Solutions to APQ) 3 3. TPJC/2007/CT/3 (a) Show that the di