Problem 12 - Applied Calculus

Question

We use \(1500 \mathrm{~kg}\) of a mineral this year and consumption of the mineral is increasing annually by \(4 \%\). The total reserves of the mineral are estimated to be \(120,000 \mathrm{~kg}\). Approximately when will the reserves run out?

Step-by-Step Solution

Answer

The reserves will run out in about 35 years.

Step 1: Understand the problem and define variables

We have an initial consumption of 1500 kg annually, which increases by 4% each year. The total mineral reserves are 120,000 kg. We need to determine the number of years it will take for consumption to reach or exceed the total reserves.

Step 2: Formulate the equation for yearly consumption

The consumption at year 0 is 1500 kg. Since consumption increases by 4% per year, consumption in year n can be modeled as:\[ C_n = 1500 \times (1.04)^n \]where \(C_n\) is the consumption in year \(n\).

Step 3: Formulate the cumulative consumption equation

The cumulative consumption over \(n\) years is given by the sum of a geometric series:\[ T_n = \sum_{k=0}^{n} 1500 \times (1.04)^k \]This simplifies to:\[ T_n = 1500 \times \frac{(1.04)^{n+1} - 1}{0.04} \]

Step 4: Set the cumulative consumption equal to total reserves

The reserves run out when the cumulative consumption equals the total reserves. Set:\[ 1500 \times \frac{(1.04)^{n+1} - 1}{0.04} = 120,000 \]

Step 5: Solve for n using logarithms

Simplify and solve for \(n\):1. Multiply both sides by 0.04:\[ 1500 \times ((1.04)^{n+1} - 1) = 4800 \]2. Divide both sides by 1500:\[ (1.04)^{n+1} - 1 = 3.2 \]3. Add 1 to both sides:\[ (1.04)^{n+1} = 4.2 \]4. Take logarithm on both sides:\[ \log((1.04)^{n+1}) = \log(4.2) \]5. Use logarithm properties:\[ (n+1) \cdot \log(1.04) = \log(4.2) \]6. Solve for \(n+1\):\[ n+1 = \frac{\log(4.2)}{\log(1.04)} \approx 35.73 \]7. Subtract 1:\[ n \approx 34.73 \]

Step 6: Approximate the result

Since \(n\) must be a whole number, approximate \(n\) to the nearest whole year. Thus, \(n \approx 35\).

Step 7: Conclusion

The reserves will run out in approximately 35 years.

Key Concepts

Geometric Series

The concept of a geometric series is fundamental when modeling growth processes like the increase in mineral consumption described in this problem. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our exercise, the initial consumption of the mineral is 1500 kg, and since the rate of increase is 4% per annum, the common ratio becomes 1.04.

A geometric series can be represented in the form:

• a, ar, ar², ar³,..., arⁿ

Since the series grows steadily, the term arises from multiplying the previous term by 1.04. This pattern of consistent multiplication perfectly describes our mineral consumption, where each year we use a certain percentage more mineral than the previous year.

Understanding geometric series helps to predict the mineral availability over time by summing the consumption each year. This sum allows us to see when the total consumption will meet or exceed the reserves.

Logarithmic Functions

In solving for when the reserves will run out, logarithmic functions play a crucial role. Logarithms are the inverse operations to exponentiation, meaning they help "undo" exponential growth and allow us to solve for the variable in an exponent.

When the cumulative consumption is set equal to the reserves, the equation stemming from the geometric series leads us to solve:

• \((1.04)^{n+1} = 4.2\)

This is where logarithms become invaluable. Instead of guessing or using lengthy trial-and-error methods, we can apply the logarithm directly:

• \[\log((1.04)^{n+1}) = \log(4.2)\]

Utilizing the property \(\log(a^b) = b\cdot\log(a)\), we simplify the process to:

• \[(n+1) \cdot \log(1.04) = \log(4.2)\]

By dividing \(\log(4.2)\) by \(\log(1.04)\), we isolate \(n+1\), allowing us to find the approximate number of years before mineral depletion will occur.

Resource Consumption Modeling

Resource consumption modeling is a strategic method used to anticipate how quickly resources will be used up. In our case, we have a finite supply of minerals, 120,000 kg, and our method explores how long these minerals will last given a progressive increase in use.

Modeling like this typically requires:

• Understanding the initial quantity used, which starts our pattern.
• Knowing the rate at which consumption increases, here, a 4% annual rise.
• Establishing the total resource amount available, here 120,000 kg.

By constructing a formula through geometric series and solving using logarithmic functions, we evaluate the future based on these assumptions. The approach may also consider external factors if the rate of growth changes or if conservation efforts alter usage habits. However, the primary model gives us a forecasting baseline, allowing businesses or governments to prepare and plan for sustainable management until alternative solutions emerge.