Problem 82 - Algebra and Trigonometry Real Mathematics, Real People

Question

You want to buy either a wood pellet stove or an electric furnace. The pellet stove costs \(\$ 3650\) and produces heat at a cost of \(\$ 19.15\) per 1 million Btu (British thermal units). The electric furnace costs \(\$ 2780\) and produces heat at a cost of \(\$ 33.25\) per 1 million Btu. (a) Write a function for the total cost \(y\) of buying the pellet stove and producing \(x\) million Btu of heat. (b) Write a function for the total cost \(y\) of buying the electric furnace and producing \(x\) million Btu of heat. (c) Use a graphing utility to graph and solve the system of equations formed by the two cost functions. (d) Solve the system of equations algebraically. (e) Interpret the results in the context of the situation.

Step-by-Step Solution

Answer

The cost functions for the pellet stove and the electric furnace are \( y = 3650 + 19.15x \) and \( y = 2780 + 33.25x \), respectively. To solve the system, set these equal and solve for \( x \). The resulting value lets you know the amount of heat at which both methods cost the same, which can help you make a cost-effective decision.

Step 1: Writing the Cost Function for the Pellet Stove

Let's denote the amount of heat produced in million BTUs as \( x \). The cost function for the pellet stove includes the initial cost (\$3650) and the cost to produce \( x \) million BTU (\$19.15 per 1 million BTU). So, the function is \( y = 3650 + 19.15x \).

Step 2: Writing the Cost Function for the Electric Furnace

Similarly, the cost function for the electric furnace includes the initial cost (\$2780) and the cost to produce \( x \) million BTU (\$33.25 per 1 million BTU). So, we have \( y = 2780 + 33.25x \).

Step 3: Graphing the Cost Functions

Plot both functions on a graph with \( y \) as the cost and \( x \) as the amount of heat produced. The intersection point, if any, represents the heat output level where both methods cost the same.

Step 4: Solving the System of Equations Algebraically

Set the two functions equal to each other and solve for \( x \): \(3650 + 19.15x = 2780 + 33.25x \).

Step 5: Interpreting the Results

The \( x \) value is the number of million BTUs where both methods cost the same. If this value is plausible in your situation (i.e. you will likely produce that amount of heat), then you can adapt your decision based on this information. The heater with the lower slope is more cost-effective in terms of operating costs and will be cheaper in the long run if you produce more than \( x \) million BTUs.

Key Concepts

Writing Cost Functions

Understanding cost functions is crucial for making informed financial decisions, especially in situations where initial investment and running costs vary. In the exercise, we explored how to write cost functions for different heating systems.

For the pellet stove with an initial cost of \(3650 and running costs of \)19.15 per million BTU, the cost function is represented as y = 3650 + 19.15x. This linear function signifies that the total cost increases by \(19.15 for each additional million BTU produced after the initial investment.

Similarly, the cost function for the electric furnace is y = 2780 + 33.25x, reflecting its initial cost of \)2780 and running costs of $33.25 per million BTU. Students should note that writing cost functions requires careful consideration of all the costs involved to make sound comparisons between different options.

Systems of Equations

Whenever we have more than one equation to solve, we are dealing with a system of equations. These equations are often related and need to be solved together to find a common solution. In the context of our exercise, the system is made up of two cost functions representing the total costs of using a pellet stove or an electric furnace.

To solve such a system, we look for the values of x (the amount of heat produced) that satisfy both equations simultaneously. This point of intersection is significant as it informs us about the scenario where both heating systems have equivalent costs. Representing cost functions as systems of equations is a fundamental skill in analyzing and comparing financial scenarios algebraically.

Graphing Utility Applications

Graphing utilities, such as calculators or software applications, are exceptional tools in visualizing algebraic equations, including cost functions. With these utilities, students can see the graphical representation of each function, making it easier to understand the relations between them.

In our heating system example, plotting both cost functions on a graph provides a clear visual indicator of where they intersect. This not only aids in understanding the system's behavior but also verifies the solutions obtained algebraically. Hands-on experience with graphing utilities helps students apply theoretical knowledge in practical, real-world scenarios.

Solving Algebraic Equations

Solving algebraic equations, such as those found in systems of cost functions, is a fundamental mathematical skill. In our exercise, the algebraic approach zeroes in on the exact point where the total costs of both heating options equal each other. By setting the two functions equal and rearranging for x, we are able to find the number of million BTUs that make the total investment and running costs identical for both options.

This method of solving algebraic equations is essential for precise and methodical decision-making processes. It reinforces the understanding of how algebra can be applied to everyday financial situations and enhances problem-solving skills.

Mathematical Interpretation

Mathematical interpretation is about making sense of the results obtained from equations and functions in relation to the real world. For instance, the solution to our system of equations indicates the breakeven point in terms of million BTUs for the pellet stove and electric furnace.

The exercise's outcome helps in understanding that the heater with the lower slope—indicating lower operating costs per million BTU—would be more cost-effective in the long run if usage exceeds the breakeven point. This illustrates how mathematical results can guide practical decision-making, and emphasizes the importance of interpreting these results within the context of the given situation.