Answer
See answers below
Work Step by Step
a) At the end of Super Bowl there are 100.000 fans. After 10 minutes there are 80000 fans left, means that $20%$ of the initial amount left the stadium. Since the rate of decrease is changing with the number of fans in the staidum, we can say there will be more than 40000 fans after 10 minutes.
b) Follow the Exponential Decay:
$$P=P_0e^{kt}$$
As $P=80000,P_0=100000,t=10$
$$80000=100000e^{10k}$$
$$\rightarrow k=-0.0223$$
Half life:
$T_{\frac{1}{2}}=-\frac{\ln 2}{k}=31.06$
c) As $t=t_0,P=15000$
$$15000=100000e^{-0.0223t}$$
Solve for $t$:
$\rightarrow t=-\frac{\ln 0.15}{0.0223} \approx 85$
d) There will be time that the stadium does not have fans. Thus, let $P=0$ and substitute to the Exponential Decay:
$$0=100000e^{-0.0223t}$$
Taking log on both sides, we get $t$ is undefined. Thus this model is not realistic from a qualitative perspective.
