Answer
(a) $104\%$
(b) $50$
(c) $n(t)=50e^{1.04t}$
(d) $5389$
(e) $6.64$ hours
Work Step by Step
(a) We use an exponential model in the form of $n(t)=n_0e^{rt}$. Use the conditions given $n(2)=400,n(6)=25600$, we have $n_0e^{2r}=400$ and $n_0e^{6r}=25600$. Take the ratio of these two equations, we have $e^{6r-2r}=25600/400$ which gives $e^{4r}=64$, thus $r=ln(64)/4\approx104\%$
(b) Plug-in the $r$ value in one of the above equations, we have $n_0e^{2\times1.04}=400$ which gives $n_0\approx50$
(c) With the above results, we can write the function as $n(t)=50e^{1.04t}$
(d) Given $t=4.5$, we have $n(4.5)=50e^{1.04\times4.5}\approx5389$
(e) Let $n(t)=50000$, we have $50e^{1.04t}=50000$ which gives $t=ln(1000)/1.04\approx6.64$ hours
