A high-altitude atmospheric research probe is launched to monitor the concentration, GGG, of a specific reactive trace gas. The rate of change of the concentration with respect to time ttt (in hours) is modeled by the differential equation
dGdt=G(16−2t)50,t≥0 \frac{dG}{dt} = \frac{G(16 - 2t)}{50}, \quad t \ge 0 dtdG=50G(16−2t),t≥0where G0G_0G0 is the initial concentration at the moment of launch (t=0t = 0t=0).
Show that the concentration is given by
G=G0e150(16t−t2)for 0≤t≤c G = G_0 e^{\frac{1}{50}(16t - t^2)} \quad \text{for } 0 \le t \le c G=G0e501(16t−t2)for 0≤t≤cwhere ccc is a constant to be found.
Find the exact maximum concentration of the gas recorded by the probe in terms of G0G_0G0. Fully justify your answer.
