Question

Oil Depletion Suppose the amount of oil pumped from one of the canyon wells in Whittier, California, decreases at the continuous rate of 10\(\%\) per year. When will the well's output fall to one-fifth of its present level?

Short answer

The well's output will fall to one-fifth of its present level in approximately 16 years.

Step-by-step solution

Identify the given values

Given the present output as 'P', the rate of decrease as \(r=0.10\) per year. The output level to which it should fall is \(\frac{1}{5}\) th of its current output. Let's label this as 'A'. To apply the decay formula \(A = Pe^{rt}\), we need to adjust the function for decay, so r will be negative, giving us the function \(A = Pe^{-rt}\).

Adjust the formula and apply the given values

Firstly, replace 'A' with \(\frac{1}{5}P\), and 'P' with 'P' in the formula, producing the equation \(\frac{1}{5}P = Pe^{-0.10t}\). 'P' on both sides of the formula cancels out, simplifying this to \(\frac{1}{5} = e^{-0.10t}\).

Isolate the variable t

To isolate 't', first apply natural log on both sides which gives \(-ln(5) = -0.10t\). Then, to completely isolate 't' divide both sides by \(-0.10\) leading to \(t = \frac{ln(5)}{0.10}\).

Solve for t

Finally, we can calculate the time as \(t = \frac{ln(5)}{0.10} = 16.1\) years approximately. So, it will take around 16 years for the well's output to fall to one-fifth of its present level.