44
Paper 1 – Rate of Change
Example
The number of bacteria , n , in a dish , after t minutes is given by
0.08t n � 1200e .
( a ) Find the value of n when t � 0.
( b ) ( i ) Find d n dt .
[ 2 ]
Solution
( ii ) Hence , write down the rate at which n is increasing when t � 24.
[ 3 ] ( c ) After k minutes , the rate of increase in n is greater than 2000 bacteria per minute .
Find the least value of k , where k � . [ 3 ]
( a )
( b ) ( i )
0.21 ( 0 ) n( 0 ) � 1200e ( M1 ) for substitution n( 0 ) � 1200 A1
|
dn
0.08t � 1200 ( e
)( 0.08 ) dt
|
( M1 ) for chain rule |
|
dn
0.08t
96e dt
|
A1
|
[ 2 ]
10
( c )
( ii )
dn dt t � k dn dt t �24 dn dt t �24
� 2000
� 654.81201 min
�1
�1
� A1 655 min
0.08
96e k � 2000 ( M1 ) for setting inequality
0.08
96e k �2000 � 0
( M1 ) for valid approach
By considering the graph of k � 37.956928 . Thus , the least value of k is 38 .
0.08k y �96e � 2000 ,
A1
[ 3 ]
[ 3 ] www . seprodstore . com
157