| Date | May 2014 | Marks available | 2 | Reference code | 14M.2.sl.TZ2.8 |
| Level | SL only | Paper | 2 | Time zone | TZ2 |
| Command term | Find | Question number | 8 | Adapted from | N/A |
Question
The number of bacteria in two colonies, \(\rm{A}\) and \(\rm{B}\), starts increasing at the same time.
The number of bacteria in colony \(\rm{A}\) after \(t\) hours is modelled by the function \(\rm{A}(t) = 12{{\text{e}}^{0.4t}}\).
Find the number of bacteria in colony \({\text{A}}\) after four hours.
Find the number of bacteria in colony \({\text{A}}\) after four hours.
How long does it take for the number of bacteria in colony \({\text{A}}\) to reach \(400\)?
The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function \(B(t) = 24{{\text{e}}^{kt}}\).
After four hours, there are \(60\) bacteria in colony \({\text{B}}\). Find the value of \(k\).
The number of bacteria in colony \({\text{B}}\) after \(t\) hours is modelled by the function \(B(t) = 24{{\text{e}}^{kt}}\).
The number of bacteria in colony \({\text{A}}\) first exceeds the number of bacteria in colony \({\text{B}}\) after \(n\) hours, where \(n \in \mathbb{Z}\). Find the value of \(n\).
Markscheme
correct substitution into formula (A1)
eg \(12{{\text{e}}^{0.4(0)}}\)
\(12\) bacteria in the dish A1 N2
[2 marks]
correct substitution into formula (A1)
eg \(12{{\text{e}}^{0.4(4)}}\)
\(59.4363\) (A1)
\(59\) bacteria in the dish (integer answer only) A1 N3
[3 marks]
correct equation (A1)
eg \(A(t) = 400,{\text{ }}12{{\text{e}}^{0.4t}} = 400\)
valid attempt to solve (M1)
eg graph, use of logs
\(8.76639\)
\(8.77\) (hours) A1 N3
[3 marks]
valid attempt to solve (M1)
eg \(n(4) = 60,{\text{ }}60 = 24{{\text{e}}^{4k}}\), use of logs
correct working (A1)
eg sketch of intersection, \(4k = \ln 2.5\)
\(k = 0.229072\)
\(k = \frac{{\ln 2.5}}{4}\) (exact), \(k = 0.229\) A1 N3
[3 marks]
METHOD 1
setting up an equation or inequality (accept any variable for \(n\)) (M1)
eg \(A(t) > B(t),{\text{ }}12{{\text{e}}^{0.4n}} = 24{{\text{e}}^{0.229n}},{\text{ }}{{\text{e}}^{0.4n}} = 2{{\text{e}}^{0.229n}}\)
correct working (A1)
eg sketch of intersection, \({{\text{e}}^{0.171n}} = 2\)
\(4.05521\) (accept \(4.05349\)) (A1)
\(n = 5\) (integer answer only) A1 N3
METHOD 2
\(A(4) = 59,{\text{ }}B(4) = 60\) (from earlier work)
\(A(5) = 88.668,{\text{ }}B(5) = 75.446\) A1A1
valid reasoning (R1)
eg \(A(4) < B(4)\) and \(A(5) > B(5)\)
\(n = 5\) (integer answer only) A1 N3
[4 marks]
