Problem 45 - Applied Calculus

Question

World consumption of tin is running at the rate of \(342 e^{0.02 t}\) thousand metric tons per year, where \(t\) is measured in years and \(t=0\) corresponds to 2014 a. Find a formula for the total amount of tin that will be consumed within \(t\) years of 2014 b. When will the known world resources of 4900 thousand metric tons of tin be exhausted? [Tin is used mainly for coating steel (a "tin" can is actually a steel can with a thin protective coating of tin to prevent rust).

Step-by-Step Solution

Answer

a. Total consumption formula: \(17100(e^{0.02 t} - 1)\). b. Tin resources exhausted in 2027.

Step 1: Understand the given function

The function given is the rate of world consumption of tin: \( 342 e^{0.02 t} \) thousand metric tons per year. Here, the consumption is dependent on \( t \), the years passed since 2014.

Step 2: Set up the integral to find total consumption

To find the total amount of tin consumed within \( t \) years, we need to integrate the consumption rate over time from 0 to \( t \). This is represented mathematically as: \[\int_{0}^{t} 342 e^{0.02 t} \, dt\]

Step 3: Integrate the function

Perform the integration of the function:\[\int_{0}^{t} 342 e^{0.02 t} \, dt = \left[ \frac{342}{0.02} e^{0.02 t} \right]_{0}^{t}\]Solve this:\[= \frac{342}{0.02}(e^{0.02 t} - 1)\]\[= 17100(e^{0.02 t} - 1)\]

Step 4: Use the total consumption formula to find when resources exhaust

We found that total consumption at time \( t \) is \( 17100(e^{0.02 t} - 1) \). Set this equal to the known tin resources to solve for \( t \):\[17100(e^{0.02 t} - 1) = 4900\]

Step 5: Solve for time \( t \) when resources will be exhausted

First, simplify the equation:\[e^{0.02 t} - 1 = \frac{4900}{17100}\]\[e^{0.02 t} = 1 + \frac{4900}{17100}\]\[e^{0.02 t} = \frac{22000}{17100}\]\[e^{0.02 t} = 1.28655\]Take the natural logarithm of both sides:\[0.02 t = \ln(1.28655)\]Calculate \( t \):\[t = \frac{\ln(1.28655)}{0.02} \approx 12.68\]Thus, the resources will be exhausted approximately 13 years after 2014.

Key Concepts

Tin Consumption

Tin consumption plays a crucial role in modern industries, particularly in the production of cans and electronic components. Tin is typically used to coat steel cans to prevent rusting, which is why these containers are commonly referred to as "tin cans."
Understanding the rate at which tin is consumed helps industries plan for sustainable use and resource management. In our exercise, the consumption rate is given as a function: \(342 e^{0.02 t}\), representing how many thousand metric tons of tin are used each year.

• This formula models tin consumption over time, starting from the year 2014.
• The exponential term \(e^{0.02 t}\) implies that tin consumption increases by a certain percentage each year.

By analyzing such patterns, industries can forecast future demand and plan for manufacturing adjustments accordingly.

Exponential Growth

Exponential growth is a powerful mathematical concept widely used to model phenomena where growth rate is proportional to the current value, leading to larger changes over time. In our case, the tin consumption is modeled with an exponential function: \(342 e^{0.02 t}\). This means the consumption rate grows by about 2% annually.

• This type of growth results in a rapidly increasing curve when plotted on a graph, illustrating how consumption accelerates over the years.
• Exponential models are common in areas such as population growth, compound interest, and natural resource consumption.

Understanding exponential growth is vital to anticipate how quickly and significantly resources like tin can deplete, emphasizing the need for sustainable usage policies.

Resource Exhaustion

Resource exhaustion occurs when a natural resource is fully depleted due to consumption outpacing replenishment or discovery of new reserves. In our example, we calculate when the known tin resources will be exhausted using the given consumption model.
First, we integrated the consumption rate to obtain a formula for total consumption over time: \(17100(e^{0.02 t} - 1)\). By setting this equal to the known tin reserves of 4900 thousand metric tons, we solved for \(t\).

• The solution involved determining when cumulative consumption surpasses the available tin, which we found to be approximately 13 years after 2014.
• This highlights how resource exhaustion computations help in planning and implementing measures to mitigate the risks of running out of critical materials.

World Resources

World resources, such as tin, are crucial for the economic and technological advancement of societies. These resources must be managed wisely to ensure long-term sustainability and availability for future generations.
The exponential model of tin consumption underscores the broader challenge of managing finite resources in an ever-expanding global economy.

• Resource exhaustion is not only about the physical depletion but also the increasing difficulty and cost to extract remaining resources.
• Countries must balance resource use with conservation efforts to prevent shortages, which necessitates global cooperation and innovative technologies.

Understanding mathematical models and resource management strategies is essential for policymakers, corporations, and individuals to make informed decisions about resource consumption and conservation.