Calculus and differential equations customs essay


121/126 Problem Solving Task Semester 2, 2013 Instructions Marks will be allocated for working and your final answer. Please ensure that you show sufficient working for the grader to follow your solution. QUESTION 1 Consider the motion of a fluid in a trough (see sketch attached). Fluid, of depth h, drains from a hole at the bottom of the trough. The motion of fluid draining through the hole is governed by Toricelli’s law, which states that the volumetric flow rate is proportional to the area of the hole and the square root of the depth of fluid. This can be written as Q = kA√ h, h ≥ 0. If the shape of the cross section of the trough is given by a parabola h(x) = x 2 (see sketch attached), the initial depth of fluid is h0, and the length of the trough is L, then: (a) Show that the volume of fluid in the trough is given by V = 4L 3 h √ h, [6 marks]. (b) Show fluid depth changes in time such that 4L 3 d(h √ h) dt = −kA√ h, [2 marks]. (c) If the area of the drainage hole is 0.1 m2 , k = 10 m1/2 /minute, L = 10 m, and the initial depth of fluid is 1 m, how long will it take for the trough to empty? [6 marks]. (d) Consider the same problem as in part (c) except now assume that additional fluid enters the top of the trough at a rate of 0.1 m3 /minute. How long does it take for the trough to empty under these conditions? [6 marks]. QUESTION 2 The Laplace transform of a function f(t) is given by F(s) = ∫ ∞ 0 e −stf(t) dt. (a) Evaluate the improper integral to show that the Laplace transform of f(t) = te at can be written as 1 (s − a) 2 , [4 marks]. (b) Use your working from part (a) to show the condition that s must satisfy so that the improper integral is convergent, [6 marks].

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