Problem 51 - College Algebra

Question

The Federal Helium Reserve held about 16 billion cubic feet of helium in 2010 and is being depleted by about 2.1 billion cubic feet each year. a. Give a linear equation for the remaining federal helium reserves, \(R,\) in terms of \(t,\) the number of years since 2010 . b. In 2015 , what will the helium reserves be? c. If the rate of depletion doesn't change, in what year will the Federal Helium Reserve be depleted?

Step-by-Step Solution

Answer

a. \( R = 16 - 2.1t \); b. 5.5 billion cubic feet in 2015; c. Reserve depletes around 2017.

Step 1: Define the Linear Equation

First, we need to define the linear equation for the remaining helium reserves. We know that the reserve starts with 16 billion cubic feet in 2010 and is being depleted by 2.1 billion cubic feet each year. This forms the equation:\[ R = 16 - 2.1t \]where \( R \) represents the remaining reserves and \( t \) is the number of years after 2010.

Step 2: Calculate Reserves for 2015

Now, let's calculate the helium reserves in 2015. Since 2015 is 5 years after 2010, we substitute \( t = 5 \) into the linear equation:\[ R = 16 - 2.1 \times 5 \]Calculate:\[ R = 16 - 10.5 = 5.5 \]So, the reserves in 2015 will be 5.5 billion cubic feet.

Step 3: Determine When Reserves Deplete

To find out when the helium reserves will be completely depleted, we need to solve for \( t \) when \( R = 0 \):\[ 0 = 16 - 2.1t \]Rearranging gives:\[ 2.1t = 16 \]\[ t = \frac{16}{2.1} \approx 7.62 \]This means the reserves will deplete approximately 7.62 years after 2010, which is during the year 2017.

Key Concepts

helium reserve

The Federal Helium Reserve is crucial because helium is a rare and non-renewable resource. It is used for a variety of important applications, such as in medical imaging, scientific research, and electronics manufacturing. In 2010, the reserve contained approximately 16 billion cubic feet of helium. Over time, this reserve is reduced due to continuous extraction and usage. This natural decline necessitates calculations to predict how long the helium will last and aids in planning future helium sourcing efforts.

rate of depletion

The rate of depletion refers to how quickly the helium reserve decreases each year. According to the exercise, this rate is 2.1 billion cubic feet annually. This means every year, the amount of helium in the reserve decreases by this fixed amount.

To comprehend the impact of this depletion rate, consider how it influences the future availability of helium:

• It directly determines the duration the reserve will last.
• Predicting depletion helps in resource management and planning for alternative solutions.

The constant rate here forms the gradient of the linear graph representing the reserve over time, offering a straightforward yet powerful insight into how resources dwindle.

solving linear equations

Solving linear equations is a critical mathematical technique used to find unknown values that satisfy a given relationship. In the context of the helium reserve exercise, the linear equation is given by:\[ R = 16 - 2.1t \]Here, \(R\) represents the remaining helium and \(t\) the number of years since 2010. This equation models a simple, linear relationship where the reserve decreases over time.

To solve linear equations:

• Identify the variables and constants.
• Apply algebraic principles to isolate the variable of interest.
• Substitute numerical values to find specific outcomes (e.g., the reserve amount in a specific year).

This process not only allows prediction—in this case of helium availability—but also enhances understanding of how linear relationships function in real-life scenarios.

real-world application of linear equations

Linear equations, like the one used to model the federal helium reserve, play a vital role in predicting future outcomes and making informed decisions. Learning how to establish and solve these equations prepares students for tackling various real-world issues. Consider these practical applications:

• Resource Management: Linear equations help in calculating how long a resource will last, enabling proactive planning.
• Budgeting: Similar principles can apply to financial forecasting.
• Engineering: Designing systems with predictable outputs often relies on linear models.

Understanding these applications highlights the importance of mathematical literacy in everyday situations, emphasizing how mastering linear equations can provide solutions to complex challenges in numerous fields.