C3 June 2009 Q3

3. Rabbits were introduced onto an island. The number of rabbits, \(P\), \(t\) years after they were introduced is modelled by the equation

\[P = 80\mathrm{e}^{\frac{1}{5}t}, \qquad t \in \mathbb{R},\ t \geqslant 0\]

(a) Write down the number of rabbits that were introduced to the island. (1)

(b) Find the number of years it would take for the number of rabbits to first exceed 1000. (2)

(d) Find \(P\) when \(\dfrac{\mathrm{d}P}{\mathrm{d}t} = 50\). (3)

Scheme	Marks
\(P = 80\mathrm{e}^{\frac{t}{5}}\)
\(t = 0 \Rightarrow P = 80\mathrm{e}^{\frac{0}{5}} = 80(1) = \underline{80}\)	B1
	(1)

Mark scheme (b)

Scheme	Marks
\(P = 1000 \Rightarrow 1000 = 80\mathrm{e}^{\frac{t}{5}} \Rightarrow \dfrac{1000}{80} = \mathrm{e}^{\frac{t}{5}}\)	M1
\(\therefore\ t = 5\ln\left(\dfrac{1000}{80}\right)\)
\(t = 12.6286\ldots\)	A1
	(2)

M1: Substitutes \(P = 1000\) and rearranges equation to make \(\mathrm{e}^{\frac{t}{5}}\) the subject.

A1: awrt 12.6 or 13 years

Note \(t = 12\) or \(t = \text{awrt } 12.6 \Rightarrow t = 12\) will score A0

Scheme	Marks
\(\dfrac{\mathrm{d}P}{\mathrm{d}t} = 16\mathrm{e}^{\frac{t}{5}}\)	M1 A1
	(2)

Mark scheme (d)

Scheme	Marks
\(50 = 16\mathrm{e}^{\frac{t}{5}}\)
\(\therefore\ t = 5\ln\left(\dfrac{50}{16}\right) \qquad \{= 5.69717\ldots\}\)	M1
\(P = 80\mathrm{e}^{\frac{1}{5}\left(5\ln\left(\frac{50}{16}\right)\right)}\) or \(P = 80\mathrm{e}^{\frac{1}{5}(5.69717\ldots)}\)	dM1
\(P = \dfrac{80(50)}{16} = \underline{250}\)	A1
	(3)
	(8 marks)

M1: Using \(50 = \frac{\mathrm{d}P}{\mathrm{d}t}\) and an attempt to solve to find the value of \(t\) or \(\frac{t}{5}\).

dM1: Substitutes their value of \(t\) back into the equation for \(P\).

A1: 250 or awrt 250