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 $n$.

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]