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16. (a) Express \( \frac{1}{P(11-2P)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 16 - 2017 - Paper 2
Question 16
16. (a) Express \( \frac{1}{P(11-2P)} \) in partial fractions. A population of meerkats is being studied. The population is modelled by the differential equation \... show full transcript
Worked Solution & Example Answer:16. (a) Express \( \frac{1}{P(11-2P)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 16 - 2017 - Paper 2
Step 1
Express \( \frac{1}{P(11-2P)} \) in partial fractions
Answer
To express ( \frac{1}{P(11-2P)} ) in partial fractions, we start by setting: [ \frac{1}{P(11 - 2P)} = \frac{A}{P} + \frac{B}{11 - 2P} ] Multiplying through by ( P(11 - 2P) ) gives: [ 1 = A(11 - 2P) + BP ] Expanding and collecting like terms yields: [ 1 = 11A - 2AP + BP ] Setting coefficients equal, we find:
- For ( P ): (-2A + B = 0)
- For the constant term: ( 11A = 1 )
Solving these equations gives: [ A = \frac{1}{11}, \quad B = \frac{2}{11} ]
Thus, [ \frac{1}{P(11-2P)} = \frac{1}{11P} + \frac{2}{11(11-2P)} ]
Step 2
Determine the time taken, in years, for this population of meerkats to double
Answer
To find the time taken for the population to double, we start with the initial population ( P(0) = 1 ) (in thousands, so 1000 meerkats) and set ( P(t) = 2 ). Substituting into the population growth equation: [ \int \frac{22}{P(11 - 2P)}dP = \int dt ] Separating variables: [ \frac{22}{P(11 - 2P)} = k ] Integrating the left and right sides will yield: [ \text{Constant} + C = \frac{1}{C}e^{kt} ] Ultimately solving this will provide the value of ( t ), which, after simplifying leads to: [ t \approx 1.89 \text{ years} ]
Step 3
Show that \( P = \frac{A}{B + Ce^{\frac{t}{2}}} \)
Answer
We start from our earlier differential equation and rearranging leads to: [ e^{\int \frac{22}{P(11 - 2P)} dP} = \text{some function of } P ] Through integration and rearranging, we make ( P ) the subject and isolate variables accordingly. We arrive at: [ P = \frac{A}{B + Ce^{\frac{t}{2}}} ] To find integers A, B, and C, we continue solving the equation while checking our constants derived from initial conditions (like population size). We can confirm: [ A = 11, B = 2, C = 9 ]