AP Calculus BC Fun 4 B Calculate Minimum and Maximum Values in Applied Contexts or Analysis of Functions Questions

An invasive species of plant appears in a fruit grove at time t=0 and begins to spread. The function C defined by $C(t)=7.6 \arctan (0.2 t)$ models the number of acres in the fruit grove affected by the species t weeks after the species appears. It can be shown that $C^{\prime}(t)=\frac{38}{25+t^{2}}$.

(Note: Your calculator should be in radian mode.)

At time t=4 weeks after the invasive species appears in the fruit grove, measures are taken to counter the spread of the species. The function A, defined by $A(t)=C(t)-\int_{4}^{t} 0.1 \cdot \ln (x) d x$, models the number of acres affected by the species over the time interval $4 \leq t \leq 36$. At what time t, for $4 \leq t \leq 36, \operatorname{does} A$ attain its maximum value? Justify your answer.