Suppose the number of rare disease cases can be reduced by 25% annually. If there are 4000 cases nationwide, how many years would it take to reduce that number of cases to 300?

To solve this problem, we can use the formula for exponential decay, which is commonly used in situations where a quantity decreases by a fixed percentage over time. The general formula is:

\[
N(t) = N_0 \times (1 - r)^t
\]

Where:
- \( N(t) \) is the number of cases after \( t \) years.
- \( N_0 \) is the initial number of cases.
- \( r \) is the annual reduction rate (as a decimal).
- \( t \) is the number of years.

Given:
- Initial number of cases \( N_0 = 4000 \)
- Reduction rate \( r = 25\% = 0.25 \)
- Final number of cases \( N(t) = 300 \)

We need to find \( t \) such that:

\[
300 = 4000 \times (1 - 0.25)^t
\]

Simplify the expression:

\[
300 = 4000 \times (0.75)^t
\]

Now, divide both sides by 4000:

\[
\frac{300}{4000} = (0.75)^t
\]

\[
0.075 = (0.75)^t
\]

To solve for \( t \), take the logarithm of both sides (using natural logarithms, but any base can be used):

\[
\ln(0.075) = t \cdot \ln(0.75)
\]

Now, solve for \( t \):

\[
t = \frac{\ln(0.075)}{\ln(0.75)}
\]

Calculate the values:

\[
t \approx \frac{-2.5903}{-0.2877} \approx 9.00
\]

So, it would take approximately 9 years to reduce the number of rare disease cases from 4000 to 300.

Would you like further details or have any questions?

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Here are 5 related questions to explore:

1. What is the general concept of exponential decay, and where else is it applied?
2. How would the time change if the reduction rate were 30% annually?
3. What happens to the number of cases if the reduction rate decreases over time instead of being constant?
4. How can you verify the accuracy of your calculation using a different method?
5. What role does logarithmic calculation play in solving exponential equations?

**Tip:** Understanding how to solve exponential decay problems is crucial in fields like finance, biology, and physics, where quantities change over time.

Learn how to solve the exponential decay problem where the number of rare disease cases decreases annually by 25%. Find out how many years it takes to reduce cases from 4000 to 300 using the exponential decay formula. Suitable for high school students studying algebra and exponential functions.

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