Edexcel IALMathematicsGeneral Applications8 Marks
Water is leaking from a cylindrical tank. The depth of the water in the tank at time t minutes is h cm. The rate at which the depth of water is changing is modelled by the differential equation dh/dt = -0.01h^(3/2). Initially, the depth of the water is 100 cm. (a) Show that the general solution of the differential equation can be written as 1/√h = 0.005t + C, where C is a constant. (b) Find the time, in minutes, when the depth of the water is 25 cm.
- (a)Show that the general solution of the differential equation can be written as 1/√h = 0.005t + C, where C is a constant.[5]
- (b)Find the time, in minutes, when the depth of the water is 25 cm.[3]
Mark scheme
Marking points from the official mark scheme. The mark or mark type is shown in brackets.
- M1 for separating variables: ∫h^(-3/2) dh = ∫-0.01 dt
- A1 for integrating h^(-3/2) to -2h^(-1/2) or -2/√h
- A1 for integrating -0.01 to -0.01t + C
- M1 for -2/√h = -0.01t + C
- A1 for 1/√h = 0.005t - C/2, leading to 1/√h = 0.005t + C' (convincing argument)
- B1 for using h=100, t=0 to find C: 1/√100 = 0.005(0) + C => C = 1/10
- M1 for substituting C = 0.1 into the equation and h=25
- A1 for t = 100 minutes
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