- (a) Express \(\frac { 3 } { ( 2 x - 1 ) ( x + 1 ) }\) in partial fractions.
When chemical \(A\) and chemical \(B\) are mixed, oxygen is produced.
A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time.
The total volume of oxygen produced, \(V \mathrm {~m} ^ { 3 } , t\) hours after the chemicals were mixed, is modelled by the differential equation
$$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 3 V } { ( 2 t - 1 ) ( t + 1 ) } \quad V \geqslant 0 \quad t \geqslant k$$
where \(k\) is a constant.
Given that exactly 2 hours after the chemicals were mixed, a total volume of \(3 \mathrm {~m} ^ { 3 }\) of oxygen had been produced,
(b) solve the differential equation to show that
$$V = \frac { 3 ( 2 t - 1 ) } { ( t + 1 ) }$$
The scientist noticed that
- there was a time delay between the chemicals being mixed and oxygen being produced
- there was a limit to the total volume of oxygen produced
Deduce from the model
(c) (i) the time delay giving your answer in minutes,
(ii) the limit giving your answer in \(\mathrm { m } ^ { 3 }\)