The population of a town was 100 000 in year 2000 and 60 000 in year 2002. The p

Question

The population of a town was 100 000 in year 2000 and 60 000 in year 2002. The population is assumed to obey the Malthusian model. (a) Find the growth constant k. (b) When will the population be 10 000? (c) According to the model, what will the population be in year 2010? (d) Write down an expression for P(t), the size of the population after years, if x - 0 in year 2010. Assume that a radioactive substance decays with a constant of decay k - 0.001 when time is measured in years. (a) How long does it take for the substance to decay from 15 grams to 3 grams? (b) If 5% of an original quantity N, is present today, when was N_0 formed initially? (c) Explain why in (b) above, we do not need to know the exact value of N_0. The populations of Country A and Country B both grow according to the Malthusian model, Country A with a doubling time 50 years and Country B with a doubling time 80 years. If the sizes of the populations were the same in year 2000, what was the ratio of the population of Country B to the population of Country A in 1950? What will the ratio be in 2050? We are given the population readings: P - 120000 in 1980. P - X in 1990, P - 500000 in 2000. What should the value of X be, so that the population grows according to a Malthusian model? Justify your answer. Assume that P(t) follows log tic growth, with initial value P_a = 100, alpha - 0.2 and b - 0.05. (a) Find P(1) and P(5). (b) Find the rate of change of P at the times t - 0, t - 1 and t - 5. (Remember that the ratio of change is the value of dP/dt!) (c) Draw the phase line of this population model. (d) Plot a sketch of the solution curve P(t) when P_0 = 100, over the interval 0 lessthanorequalto t lessthanorequalto 6. Use the information in your answers to (a), (b) and (c) to make the sketch as accurate as possible. For each of the following systems, draw a sketch of: (i) dx/dt as a function x, (ii) the phase line. Also, for each system, list all the equilibrium points and classify each of them as stable or unstable: and predict the outcome of the solution if the system starts at x(0) - 0.5. (a) dx/dt = 2 - x (b) d/dt = (x - 1) x^2 (c) dx/dt = (x - 2)x (x + 1) A tank initially contains 200 litres of liquid A. Liquid B is pumped into the tank at the rate of 10 litres per minute. The mixture is stirred continuously and the tank is kept full at 200 litres at all times. The contents of the tank is pumped out at the same rate of 10 litres per minute. Let X (t) denote the amount (in litres) of liquid B in the tank after t minutes. (a) Write down the differential equation for X(t), and its initial value. (b) Solve the initial value problem in (a) to find the solution X(t) or all t. (c) How long does it take until the tank contains the same amount of both liquid A and liquid B?

Explanation / Answer

You have asked way too many questions, only 1 question is allowed per chegg question.

I am answering the second one for you.

(2)

Given:

k = 0.001

(a)

Using the first order equation for radioactive decay:

ln(N0/N) = k*t

Putting values:

ln(15/3) = 0.001*t

Solving we get:

t = 1609.44 years

(b)

Given:

N = 0.05*N0

Putting values:

ln(1/0.05) = 0.001*t

Solving we get:

t = 2995.73 years ago

(c)

This is because N is expressed as a percentage of N0, and inside the 'ln' term in the formula, the N0 will get cancelled because of the ratio N/N0. Hence the exact value of N0 is not required.

Hope this helps !

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