The output of a particular producing oil well decreases at the continuous rate of 20​% per year. How long it will take for the output to fall to one-eighth of its present​ output?

To solve this problem, we can use the exponential decay formula:

\[
Q(t) = Q_0 \cdot e^{-rt}
\]

Where:
- \( Q(t) \) is the output at time \( t \),
- \( Q_0 \) is the initial output,
- \( r \) is the rate of decay (in this case 20% or 0.20),
- \( t \) is the time in years,
- \( e \) is Euler's number (approximately 2.71828).

We are given that the output falls to one-eighth of its original value, so \( Q(t) = \frac{1}{8} Q_0 \). Substituting this into the equation:

\[
\frac{1}{8} Q_0 = Q_0 \cdot e^{-0.20t}
\]

We can divide both sides by \( Q_0 \):

\[
\frac{1}{8} = e^{-0.20t}
\]

Now, take the natural logarithm of both sides to eliminate the exponential term:

\[
\ln\left(\frac{1}{8}\right) = -0.20t
\]

Using the property of logarithms:

\[
\ln(1) - \ln(8) = -0.20t
\]

Since \( \ln(1) = 0 \) and \( \ln(8) \approx 2.079 \), the equation becomes:

\[
-2.079 = -0.20t
\]

Now, solve for \( t \):

\[
t = \frac{2.079}{0.20} \approx 10.395 \, \text{years}
\]

Therefore, it will take approximately 10.4 years for the output to fall to one-eighth of its present value.

Would you like more details or have any other questions?

Here are 5 related questions to expand your understanding:
1. What is the exponential growth formula and how does it differ from decay?
2. How would the time change if the decay rate were 15% instead of 20%?
3. How can logarithmic functions be used in other real-world applications?
4. What happens to the decay rate if the process isn’t continuous but happens at regular intervals?
5. How does compound interest compare to continuous growth/decay?

**Tip:** Natural logarithms are key when working with continuous growth or decay, as they help simplify exponential equations.

The output of a particular producing oil well decreases at the continuous rate of 20​% per year. How long will it take for the output to fall to one-eighth of its present​ output?

Explore how to calculate the time it takes for an oil well's output to decrease to one-eighth of its original value using the exponential decay formula. This problem covers key concepts in exponential decay and logarithms, suitable for students in Grades 11-12.

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