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16. (a) Express \( \frac{1}{P(11 - 2P)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 2 - 2017 - Paper 2
Question 2
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 equat... show full transcript
Worked Solution & Example Answer:16. (a) Express \( \frac{1}{P(11 - 2P)} \) in partial fractions - Edexcel - A-Level Maths Pure - Question 2 - 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 write it as:
[ \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 ]
Expanding this yields:
[ 1 = 11A - 2AP + BP ]
This implies: [ (B - 2A)P + 11A = 1 ]
From this equation, we can set up two equations:
- ( B - 2A = 0 )
- ( 11A = 1 )
Solving these, we find ( A = \frac{1}{11} ) and substituting this into the first equation gives ( 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
The initial population when the study began is 1000 meerkats, or ( P(0)=1 ) in thousands. Doubling this gives ( P(t) = 2 ). Using the differential equation, we separate variables:
[ \int \frac{22}{P(11 - 2P)} dP = \int dt ]
Integrating both sides gives:
[ 22 \ln |P| + 22 \ln |11 - 2P| = t + C ]
To find time ( t ), we substitute ( P = 1 ) when ( t = 0 ) to find constant ( C ), yielding:
After some calculations, we can determine ( t ) using the known value of ( P = 2 ), leading to a time of approximately 1.89 years before the population doubles.
Step 3
Show that \( P = \frac{A}{B + Ce^{\frac{1}{2}t}} \).
Answer
Using the laws of logarithms on the previous equation:
[ 2 \ln P - 2 \ln(11 - 2P) = t - C ]
We can exponentiate both sides:
[ \frac{P}{11 - 2P} = e^{\frac{1}{2}t} ]
Rearranging gives:
[ 9P = (11 - 2P)e^{\frac{1}{2}t} ]
Thus, simplifying leads to:
[ P = \frac{A}{B + Ce^{\frac{1}{2}t}} ]
Here, we find substitutions for ( A ), ( B ), and ( C ) to be integers based on our earlier calculations, which verifies the form as required.