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# MATH1400: Modelling with dierential equations, 2012-13 Examples 3 Dr. Teed, 8.18c, Department of Applied Mathematics. Please do section 1 during the...

please take note on the attachment. I need help on section 2 question 1,2 and 5. please, thank you.

MATH1400: Modelling with diﬀerential equations, 2012–13 Examples 3 Dr. R. J. Teed, 8.18c, Department of Applied Mathematics. Please do section 1 during the tutorials on 1st and 8th March. There will also be a quiz on 1st March . The topics for the quiz will be announced during lectures. Hand in your answers to section 2 in the grey boxes outside the School of Maths Taught Student Oﬃce by 5pm on Monday 11th March . Be sure to put your work in your tutor’s box. Be sure to write your name on your work. Your work will be returned during tutorials. In all cases, clearly specify the ODE that must be solved, along with its initial condition, and use the appropriate method to ﬁnd the particular solution.that is independent of the choice of name or coordinate system you used to solve the problem. Section 1: to be covered in tutorials. 1 . A nuclear breeder reactor produces waste that contains (amongst other things) the isotope 239 Pu (Plutonium-239). After 15 years, the initial concentration of 239 Pu in the waste has decreased by 0.043%. Find the half-life of the isotope. 2 . A particle is acted on by an oscillating force; there is also an air resistance force that opposes the motion. The speed v of the particle is governed by the ODE dv dt = F sin t - kv, where F is proportional to the strength of the oscillating force and k is a constant giving the size of the air resistance. What kind of ODE is this? Suppose that at time t = 0, the speed is v 0 . Find v ( t ). 3 . Solid cancerous tumours do not grow exponentially in time. As the tumour becomes larger, the doubling time increases. Research has shown that the number of cells n ( t ) in a tumour at time t satisﬁes the equation n ( t ) = n 0 exp ± λ α ( 1 - e - αt ) ² , where n 0 , λ and α are constants. (a) Show that n satisﬁes the diﬀerential equation dn dt = λe - αt n, with n (0) = n 0 . (b) A tumour satisﬁes the above equation with α = 0 . 02. Originally, when it contained 10 4 cells, the tumour was increasing at a rate of 20% per unit time. What is the limiting number of cells in the tumour? 4 . A body is found at the scene of a crime at 6 a.m. – at that time, the temperature of the body was 31 C. By 8 a.m., the temperature had cooled to 29 C. The ambient temperature was 20 C, and normal body temperature is 37 C. Estimate when the murder occurred. 1
5 . Assume that Lake Erie has a volume V of 480 km 3 of water and that its rate of inﬂow r i (from Lake Huron) and outﬂow r o (to Lake Ontario) are both 350 km 3 per year. Suppose that at the time t = 0 (years), the pollutant concentration of Lake Erie (caused by past industrial pollution) is ﬁve times that of Lake Huron, and that the concentration of pollution, c i , in Lake Huron does not change. Assume that the outﬂow from Lake Erie is perfectly mixed lake water. Show that we can model the amount of pollution, x ( t ), in Lake Erie by dx dt = r i c i - ± r o V ² x How long will it take to reduce the pollution concentration in Lake Erie to the level of twice that of Lake Huron? Section 2: to be handed in 1 . A small piece of the Turin Shroud was subjected to Carbon dating in 1988. It was found that 92 . 3% of the original 14 C remained. Calculate (to the nearest year) the year that this suggests the shroud was made. (In 2005, it was suggested that the 1988 sample was taken from an area where the original shroud might have been repaired.) 2 . The population p ( t ) of the county of Midsommershire obeys the Logistic equation: dp dt = 0 . 04 p ± 1 - p 10 6 ² where t is measured in years. (a) Modify the equation to take into account the fact that 10 000 people move away from Midsommershire each year. Find the general solution p ( t ). (b) If the population was 8 × 10 6 in 1970, ﬁnd the particular solution. What happens as t → ∞ ? 3 . Suppose that the ambient temperature varies sinusoidally in time. Then Newton’s Law of cooling becomes: d Θ dt = - k - A sin( ωt )) , with Θ(0) = Θ 0 . Solve this equation, and describe the solution in the limit t → ∞ . 4 . As the salt KNO 3 dissolves in methanol, the number x ( t ) of grams of the salt in a solution after t seconds satisﬁes the diﬀerential equation dx dt = 0 . 8 x - 0 . 004 x 2 . (a) What is the maximum amount of the salt that will ever dissolve in the methanol? (b) If x = 50 when t = 0, how long will it take for an additional 50g of salt to dissolve? 5 . A lecture theatre contains a volume of air V = 120 000 m 3 . It is ventilated by a system that pumps in fresh air at a rate r , and that draws out stale air at the same rate. Assume that the air in the theatre is mixed well. The ventilation system is required to be able to reduce any impurities to a level of 1% of their initial concentration in 30 min. Find the required pumping rate r (in m 3 / min). You may leave your answer in terms of logs, or use a calculator. 2

final answer for problem 3. Derivation same as previous... View the full answer

Monday, March 11, 2013
10:40 PM Unfiled Notes Page 1

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