An industrial chemist is monitoring the concentration of a catalyst, $C$ mg/L, in a pressurized chemical reactor. The concentration at time $t$ minutes after the reaction begins is modelled by the equation

$$C=\frac{150\lambda e^{0.15t}}{\lambda e^{0.15t}+12}$$

where $\lambda$ is a positive constant.

Given that:

• The initial concentration of the catalyst at the start of the process was $30$ mg/L.
• The concentration reached $90$ mg/L after $T$ minutes.