Experts claim there is enough coal left in the world for at least another 400 yrs at current rates of consumption. Estimate how long the coal would last if its annual usage rate increases by 5% and by 10% per year.

To estimate how long the coal reserves would last with annual consumption increasing by 5% and 10%, we can model the problem using exponential growth. The general formula for consumption growing at a constant rate \(r\) is:

\[
C(t) = C_0(1 + r)^t
\]

Where:
- \(C(t)\) is the total coal consumed after \(t\) years,
- \(C_0\) is the initial annual consumption,
- \(r\) is the growth rate (in decimal form, i.e., 0.05 for 5% and 0.10 for 10%),
- \(t\) is the number of years.

However, since we want to calculate the time it would take for the coal reserves to be depleted, we need to use a different approach. We are given:
- \(R_0 = 400 \times C_0\), where \(R_0\) is the total amount of coal remaining (assuming 400 years' worth at current consumption),
- We need to find how long the coal lasts if consumption increases at a rate \(r\).

The depletion time \(T\) can be calculated using the formula for the sum of a geometric progression since the annual consumption increases geometrically. The total amount consumed over \(T\) years with a growth rate \(r\) is given by:

\[
R_0 = C_0 \times \frac{(1 + r)^T - 1}{r}
\]

Solving for \(T\):

\[
T = \frac{\log\left(1 + \frac{R_0 \times r}{C_0}\right)}{\log(1 + r)}
\]

Since \(R_0 = 400 \times C_0\), the formula simplifies to:

\[
T = \frac{\log\left(1 + 400r\right)}{\log(1 + r)}
\]

Let's calculate \(T\) for both 5% (\(r = 0.05\)) and 10% (\(r = 0.10\)) growth rates. I'll compute the results next.If the annual coal consumption increases by:
- 5% per year, the coal reserves would last approximately **62.4 years**.
- 10% per year, the coal reserves would last approximately **39 years**.

These numbers show that a significant increase in the consumption rate dramatically reduces the lifespan of the coal reserves.

Would you like further clarification on any of these calculations or concepts? 

Here are five related questions for further exploration:
1. How would coal consumption change if the growth rate was reduced to 2% annually?
2. How does a compounding consumption rate differ from a constant rate?
3. What factors could lead to a reduction in coal usage growth rates?
4. How can technological advancements impact coal depletion?
5. What are the environmental implications of accelerating coal consumption?

**Tip:** In exponential growth scenarios, even small changes in the growth rate can have a major impact over time!

Calculate how long coal reserves will last if the annual usage rate increases by 5% or 10% per year. This problem involves exponential growth, geometric progressions, and logarithmic calculations, suitable for students in Grades 11-12.

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