3 In a chemical process, the mass \(\boldsymbol { M }\) grams of a chemical at time \(\boldsymbol { t }\) minutes is modelled by the differential equation
$$\underset { d t } { d \underline { \underline { M } } } - \underset { \mathrm { z } \left( \left. \right| ^ { \prime } + \mathrm { z } ^ { 2 } \right) ^ { \prime } } { M _ { - } }$$
- Find \({ } _ { \mathrm { f } } \overline { 1 } ; \overline { 12 } d t\)
(li) Find constants \(\boldsymbol { A } , \boldsymbol { B }\) and \(\boldsymbol { C }\) such lhat
$$\begin{array} { c c }
1 & B t + C
\hdashline t \left( I + t ^ { 2 } \right) & I
\end{array} . + \begin{array} { c c }
I & I + 1 ^ { 2 } .
\end{array}$$ - Use integration, together with your results in parts (i) and (ii), to show that
$$M = \frac { K t } { J \sqrt { + , 2 } } ,$$
where \(\boldsymbol { K }\) is a constant.
- When \(\boldsymbol { t } = \mathrm { I } , \boldsymbol { M } = 25\). Calculate \(\boldsymbol { K }\)
What is the mass of the chemical in the long term?