We need to find a function f(x) as shown below such that it models the oil spill covering,
[tex]\begin{gathered} f(t)=ab^t \\ a,b\rightarrow\text{ constants} \end{gathered}[/tex]Therefore, in our case, since the oil spill initially covers 2mi^m,
[tex]\begin{gathered} f(0)=ab^0=a*1=a \\ and \\ f(0)=2 \\ \Rightarrow a=2 \end{gathered}[/tex]On the other hand, after 4 hours the area triples; therefore, at t=4, the covered area is 3*2=6mi^2. Use this fact to fnd bthe value of b, as shown below
[tex]\begin{gathered} f(4)=2b^4 \\ and \\ f(4)=6 \\ \Rightarrow2b^4=6 \\ \Rightarrow b=\sqrt[4]{3}=(3)^{\frac{1}{4}} \end{gathered}[/tex]Thus, the exponential model is
[tex]\Rightarrow A(t)=2(3)^{\frac{t}{4}}[/tex]