Problem 51 - College Algebra with Corequisite Support

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. (c) Year 2018.

Step 1: Define the Linear Equation

To find the linear equation for the remaining helium reserves, recognize the situation as a linear relationship because of a constant rate of depletion. Initially, in 2010, the reserves were 16 billion cubic feet. With each subsequent year, 2.1 billion cubic feet are depleted. Define the reserves, \( R \), in terms of the number of years since 2010, \( t \). The equation is formulated as \( R = 16 - 2.1t \). This equation's structure is \( R = R_0 - rt \), where \( R_0 \) is the initial reserve (16 billion), and \( r \) is the depletion rate (2.1 billion cubic feet per year).

Step 2: Calculate Reserves in 2015

To find the helium reserves in 2015, substitute \( t = 5 \) (because 2015 is 5 years after 2010) into the equation \( R = 16 - 2.1t \). Perform the calculation: \( R = 16 - 2.1 \times 5 \). Simplifying, \( R = 16 - 10.5 = 5.5 \). Thus, the reserves in 2015 will be 5.5 billion cubic feet.

Step 3: Determine When Reserves Are Depleted

To find when the reserves will be depleted, set \( R = 0 \) in the equation \( R = 16 - 2.1t \) because depletion is complete when reserves are zero. Solve for \( t \): \( 0 = 16 - 2.1t \). Rearrange to \( 2.1t = 16 \), then simplify to find \( t = \frac{16}{2.1} \approx 7.62 \). Since \( t \) is the number of years after 2010, round 7.62 to 8 years for a practicality check, leading to complete depletion in the year 2018 (2010 + 8).

Key Concepts

Rate of Change

Rate of change is an important mathematical concept that refers to how a quantity changes over time. In problems involving linear equations, like the helium reserve scenario, the rate of change is constant, meaning the quantity changes at the same rate every time period. In this scenario, every year, the helium reserve decreases by 2.1 billion cubic feet, which represents the constant rate of change.
Understanding this allows us to model the situation with a linear equation, which is a straight line when graphed. The constant rate of change forms the slope of this line.

• The initial amount of helium in 2010 is 16 billion cubic feet, which is called the "y-intercept" in graph terms.
• The rate of change, or slope, is -2.1 billion cubic feet per year because the reserves are decreasing.
• Each year that passes, the total amount of helium decreases by the same rate.

This helps predict how much helium will remain in future years by continuing the pattern established by the rate of change.

Depletion Problems

Depletion problems are scenarios where a resource diminishes over time until it is completely gone, making them excellent for applying linear equations. In our helium reserve example, the depletion is constant at 2.1 billion cubic feet per year. This is a straightforward calculation in algebra that allows us to predict future or past values based on the constant depletion rate.
These problems often require students to determine when the reserves will reach a specific value. Here, the goal was to find the year when the reserve would become zero. This involves setting the linear equation to zero and solving for the number of years.

• Set the remaining reserve equation to zero, as depletion problems often aim to find when resources are fully exhausted.
• The equation involves solving for time, a classic problem in depletion scenarios.
• Calculate how many years it takes through simple algebraic manipulations.

Depletion problems like this help in understanding how diminishing resources behave over time, which can be applied to various real-world situations beyond just helium reserves.

Algebraic Modeling

Algebraic modeling is the process of creating equations to represent real-world situations. This allows complex scenarios or data to be simplified and analyzed mathematically, such as predicting future trends. In the helium reserve problem, algebraic modeling involves translating the facts into a linear equation.
The initial amount of reserves and the rate of depletion are modeled in the equation: \[ R = 16 - 2.1t \]
Steps for algebraic modeling include:

• Identify the constant rate or quantity per time period, which in this case is the depletion rate of helium.
• Recognize the initial starting point, here 16 billion cubic feet in 2010.
• Construct an equation that ties these real-world elements together to forecast changes over time.

This type of modeling is not only applicable for educational purposes but is widely used in fields like economics, engineering, and environmental science to solve practical problems. By mastering algebraic modeling, one can effectively interpret and shape predictions based on linear relationships.