Suppose the amount of oil pumped from one of the wells

Question

Suppose the amount of oil pumped from one of the wells in Whittier, California, decreases at the continuous rate of 8% per year. When to the nearest year, will the well's output falll to one-eight of its present value?

Answer 1

Let the present value be x.

Decreasing at 8%, next year it will be x(1-0.08). This is the present value decreased by 8%

The following (second) year, it will be x(1-0.08)(1-0.08) = x(1-0.08)^2 where the amount from above decreases by another 8%

So, over a certain number of years (say, n) the amount left will be x(1-0.08)^n

In our problem, we want to know after how many years, the amount left will be 1/8th the original amount (x/8). Mathematically, we need to solve: x(1-0.08)^n = x/8 for n. We see that x drops out (cancels) and we are left with

0.92^n = 1/8

Take the log of both sides:

log(0.92)^n = log(1/8)

using the properties of logarithms:

n*log(0.92) = log(1) - log(8)

log(1) = 0 so,

n = -log(8)/log(0.92) = 24.9

Rounding to the nearest year, n = 25 years.

Answer 2

How much is in the tank to begin with?