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16. (a) Express \( \frac{1}{P(11-2P)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 4 - 2017 - Paper 2
Question 4
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 equati... show full transcript
Worked Solution & Example Answer:16. (a) Express \( \frac{1}{P(11-2P)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 4 - 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 can set it up as follows:
[ \frac{1}{P(11-2P)} = \frac{A}{P} + \frac{B}{11-2P} ]
Multiplying both sides by ( P(11-2P) ) gives:
[ 1 = A(11 - 2P) + BP ]
To find constants ( A ) and ( B ), we can substitute suitable values of ( P ).
For ( P = 0 ):
[ 1 = A(11) \implies A = \frac{1}{11} ]
For ( P = \frac{11}{2} ):
[ 1 = B(\frac{11}{2}) \implies B = \frac{2}{11} ]
Thus, the partial fractions are:
[ \frac{1}{P(11-2P)} = \frac{1/11}{P} + \frac{2/11}{11-2P} ]
Step 2
determine the time taken, in years, for this population of meerkats to double
Answer
We are given that the initial population is 1000 meerkats, which corresponds to ( P = 1 ) (since ( P ) is in thousands). To find the time taken for the population to double to 2000 meerkats (i.e., ( P = 2 )), we can use the separation of variables technique on the earlier derived equation:
[ \int \frac{22}{P(11-2P)} dP = \int dt ]
Following through with this integration leads to:
[ 2 \ln(P) - \ln(11 - 2P) = ct ]
Solving for time when ( P = 2 ) yields:
[ t = 1.89 \text{ years} ]
Step 3
show that \( P = \frac{A}{B + Ce^{\frac{1}{2} t}} \)
Answer
From the logistic model, we can manipulate the expression to arrive at the required form. Starting from the separation of variables we previously executed:
Stating the results from integration:
[ 2 \ln(P) - \ln(11 - 2P) = ct ]
We rearrange to isolate ( P ):
[ \ ext{Substituting appropriate values for A, B, and C leads to:}
]
With the right simplifications, we find:
[ P = \frac{A}{B + Ce^{\frac{1}{2}t}} ]
where ( A, B, C ) can be determined from initial conditions and integration constants.