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Chapter 11: Problem 11

At the end of 2008 , the total reserve of a mineral was \(350,000 \mathrm{~m}^{3}\). In the year 2009 , about \(5000 \mathrm{~m}^{3}\) was used. Each year, consumption of the mineral is expected to increase by \(8 \%\). Under these assumptions, in how many years will all reserves of the mineral be depleted?

Short Answer

Expert verified
The reserves will be depleted in 38 years after 2008.

Step by step solution

01

Calculate the consumption for the first year

Given that in 2009, the consumption was 5000 m鲁. Each subsequent year, this consumption increases by 8%. Therefore, we start with 5000 m鲁 of consumption in 2009.
02

Create the consumption sequence

We need to create a sequence for the annual consumption: 2009 has 5000 m鲁, 2010 will have an increased consumption by 8%, and so on.For any given year, the consumption can be calculated as:\[ C_n = 5000 \times (1.08)^{n-1} \]where \(C_n\) is the consumption for the \(n\)-th year after 2008.
03

Determine the number of years for depletion

We set up the inequality to find the year \(N\) when the total reserved amount of 350,000 m鲁 is used up:\[ \sum_{n=1}^{N} C_n = 350,000 \]Using the formula for the sum of a geometric series, we have:\[ \sum_{n=1}^{N} C_n = 5000 \times \frac{(1.08)^N - 1}{1.08 - 1} \]This must be equal to 350,000:\[ 5000 \times \frac{(1.08)^N - 1}{0.08} = 350,000 \]
04

Solve the equation for N

First, rearrange the equation:\[ 625 \times ((1.08)^N - 1) = 350,000 \]\[ (1.08)^N - 1 = 560 \]\[ (1.08)^N = 561 \]Apply logarithms to solve for \(N\):\[ N \times \log(1.08) = \log(561) \]\[ N = \frac{\log(561)}{\log(1.08)} \]Calculating gives \(N \approx 37.35\).
05

Conclusion on the number of years

Since \(N\) must be a whole number, and reserve depletion occurs on overlapping over \(37.35\) years, we round up to the nearest whole number, which means the reserves will be depleted in 38 years after 2008.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Consumption Growth
When we talk about **consumption growth** in this context, we are referring to the increase in the amount of a mineral that is consumed annually. In the exercise, consumption starts at 5000 cubic meters in 2009 and is expected to grow by 8% each year.

Consumption growth is a common consideration in resource management and economics.
  • The initial amount is also known as the base consumption value, which is 5000 m鲁 in this case.
  • Growth rate is expressed as a percentage鈥攊n this example, 8%.
  • To determine the consumption for each subsequent year, multiply the previous year's consumption by the growth rate factor, which is 1.08 (reflecting the original amount and the 8% increase).
This escalating figure is vital, as it indicates increased demand or usage over time. As each year passes, the amount consumed increases, meaning that reserves will deplete faster than if consumption were constant.
Understanding Reserve Depletion
**Reserve depletion** occurs when the available quantity of a resource, like the mineral in this exercise, diminishes to exhaustion. Given the initial reserve of 350,000 m鲁, we need to project the decreasing amount based on the growing consumption rate.
  • Every year, the total reserve is reduced by the amount consumed during that year.
  • The depletion continues until the total consumption equals the total initial reserves.
  • This model assumes no additional reserves are discovered or added.
In the context of resource management, understanding reserve depletion helps strategize future use and conservation efforts. It provides vital insights into when alternative resources might need to be explored or other contingency plans developed. In this exercise, we calculated how long it would take until all reserves are gone given the ever-increasing rate of consumption.
Logarithmic Calculations Simplified
**Logarithmic calculations** often seem intimidating, but they're quite helpful in solving exponential problems like this one. Let's break it down in the context of determining reserve depletion.
  • Logarithms help "reverse" exponential calculations to find unknown variables鈥攍ike the exact number of years before reserves run out.
  • In our equation, we used the formula \[ (1.08)^N = 561 \] to express the problem. Here, the logarithm helps isolate \(N\), the unknown number of years.
  • You achieve this by taking the log of both sides, allowing for the equation to be rewritten as: \[ N \times \log(1.08) = \log(561) \]
This operation changes a problem that seems complex into a straightforward division of logarithms: \[ N = \frac{\log(561)}{\log(1.08)} \]
Finally, remember that logarithms also follow special rules, simplifying how we deal with rapid growth scenarios like in our mineral reserve problem. By applying these rules, we determine the precise timeline for when the reserves will last.

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