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7. (a) Express \( \frac{2}{P(P - 2)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 8 - 2015 - Paper 4
Question 8
7. (a) Express \( \frac{2}{P(P - 2)} \) in partial fractions. A team of biologists is studying a population of a particular species of animal. The population is mo... show full transcript
Worked Solution & Example Answer:7. (a) Express \( \frac{2}{P(P - 2)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 8 - 2015 - Paper 4
Step 1
Express \( \frac{2}{P(P - 2)} \) in partial fractions.
Answer
To express ( \frac{2}{P(P - 2)} ) in partial fractions, we start by setting:
[ \frac{2}{P(P - 2)} = \frac{A}{P} + \frac{B}{P - 2} ]
Multiplying through by the common denominator ( P(P - 2) ) gives:
[ 2 = A(P - 2) + BP ]
To find ( A ) and ( B ), we can select suitable values for ( P ) to simplify.
Let ( P = 0 ): [ 2 = A(0 - 2) + 0 \Rightarrow A = -1 ]
Let ( P = 2 ): [ 2 = 0 + B(2) \Rightarrow B = 1 ]
Thus, we have:
[ \frac{2}{P(P - 2)} = \frac{-1}{P} + \frac{1}{P - 2} ]
Step 2
solve this differential equation to show that \( P = \frac{6}{3 - e^{-2\sin 2t}} \)
Answer
Starting with the differential equation:
[ \frac{dP}{dt} = \frac{1}{2} P(P - 2) \cos 2t ]
We separate variables:
[ \frac{1}{P(P - 2)} dP = \frac{1}{2} \cos 2t , dt ]
Integrating both sides gives:
[ \int \frac{1}{P(P - 2)} dP = \frac{1}{2} \int \cos 2t , dt ]
The left side integrates to:
[ \ln |P - 2| - \ln |P| = \sin 2t + C ]
Combining the logs: [ \ln \left( \frac{P - 2}{P} \right) = \sin 2t + C ]
Exponentiating: [ \frac{P - 2}{P} = e^{\sin 2t + C} \Rightarrow \frac{P - 2}{P} = e^C e^{\sin 2t} ]
Setting ( P = 3 ) when ( t = 0 ) helps us find ( C ): [ \frac{1}{3} = e^C \Rightarrow C = \ln 3 ]
Thus, we can express the equation as: [ P = \frac{6}{3 - e^{-2\sin 2t}} ]
Step 3
find the time taken for the population to reach 4000 for the first time.
Answer
To find when the population reaches 4000, we set:
[ P = 4000 \Rightarrow \frac{6}{3 - e^{-2\sin 2t}} = 4000 ]
Rearranging gives: [ 3 - e^{-2\sin 2t} = \frac{6}{4000} \Rightarrow e^{-2\sin 2t} = 3 - \frac{3}{2000} ]
Solving for ( e^{-2\sin 2t} ): [ e^{-2\sin 2t} = \frac{5997}{2000} ]
Taking the natural logarithm: [ -2\sin 2t = \ln \left( \frac{5997}{2000} \right) ]
Next, solving for ( , t ): [ \sin 2t = -\frac{1}{2} \ln \left( \frac{5997}{2000} \right) ]
Finding ( 2t ) and then dividing by 2 gives the time in years. Finally, rounding to three significant figures will yield the answer.