Question

My first observed oil spill covers 2 mi.² measurement so that the area is tripling every four hours find an exponential model the area A (in mi^2 of the oil spill function of time t in hour from beginning of the spill

Answer 1

We need to find a function f(x) as shown below such that it models the oil spill covering,

`\begin{gathered} f(t)=ab^t \\ a,b\rightarrow\text{ constants} \end{gathered}`

Therefore, in our case, since the oil spill initially covers 2mi^m,

`\begin{gathered} f(0)=ab^0=a*1=a \\ and \\ f(0)=2 \\ \Rightarrow a=2 \end{gathered}`

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

`\begin{gathered} f(4)=2b^4 \\ and \\ f(4)=6 \\ \Rightarrow2b^4=6 \\ \Rightarrow b=\sqrt[4]{3}=(3)^{(1)/(4)} \end{gathered}`

Thus, the exponential model is

`\Rightarrow A(t)=2(3)^{(t)/(4)}`

The answer is A(t)=2(3)^(t/4); A(t) is in i^2 ,and t is in hours.