Problem 46 - Algebra 2

Question

If you recycle a 3\(\frac{1}{2}\) foot stack of newspapers, one less 20-foot loblolly pine tree will be needed for paper. Use a prediction equation to determine how many feet of loblolly pine trees will not be needed for paper if you recycle a pile of newspapers 20 feet tall.

Step-by-Step Solution

Answer

Recycling a 20-foot stack of newspapers saves about 114.29 feet of trees.

Step 1: Understanding the Problem

First, we need to understand the problem. We are given that a 3\(\frac{1}{2}\) foot stack of newspapers saves one 20-foot tree. The goal is to predict how many feet of trees we will save if we recycle a 20-foot stack of newspapers.

Step 2: Setting Up the Proportion

To find out how many 20-foot trees are saved, we set up a proportion based on the given information. We know that:\[\frac{3.5}{20} = \frac{20}{x}\]where 3.5 is the height of the newspaper stack that saves one tree, and 20 is the height of the new stack we want to analyze.

Step 3: Solving for the Unknown

Now we solve the proportion for \(x\), which is the total feet of trees saved:Cross multiply to get the equation:\[3.5x = 400\]

Step 4: Finding the Solution

Divide both sides by 3.5 to solve for \(x\):\[x = \frac{400}{3.5}\]Compute \(x\):\[x \approx 114.29\]

Step 5: Conclusion

If you recycle a 20-foot stack of newspapers, approximately 114.29 feet of loblolly pine trees will not be needed.

Key Concepts

Problem Solving

Problem solving in mathematics often involves understanding the situation thoroughly before diving into calculations. In this exercise, we are tasked with determining how many feet of loblolly pine trees can be saved by recycling a 20-foot stack of newspapers.
Understanding the problem is the first critical step. We know that recycling a 3.5-foot stack saves a 20-foot tree. This serves as our key piece of information.
Once the problem is understood, it's crucial to decide on a method to arrive at a solution. Here, setting up a proportion is the chosen method, as it allows us to relate different quantities in a mathematical way that is easy to calculate. Problem-solving like this often involves breaking down the problem into smaller, more manageable parts and using these to build up to the final solution.

Prediction Equation

A prediction equation is a mathematical way to forecast an unknown quantity based on already known quantities. In this problem, we wanted to predict how many feet of trees are saved by recycling.
We set up a proportion where one known relationship (3.5 feet of newspapers saves a 20-foot tree) is used to predict another unknown relationship about the 20-foot newspaper stack.
By setting up the equation \(\frac{3.5}{20} = \frac{20}{x}\), we use known quantities to solve for \(x\), which represents the feet of trees saved. This equation is central to translating the real-world conservation question into a solvable mathematical format.

Algebraic Manipulation

Algebraic manipulation is vital in solving equations and proportions efficiently. It involves using mathematical operations to transform equations into simpler forms.
In this exercise, to solve \(\frac{3.5}{20} = \frac{20}{x}\), cross multiplication is used. This technique involves multiplying across the equals sign diagonally: \(3.5x = 400\).
Once in this form, the equation is simpler to solve algebraically by isolating \(x\). Dividing both sides by 3.5 focuses on solving for \(x\), ultimately finding the solution: \(x \approx 114.29\).

• Understand cross-multiplication as a method to eliminate fractions in proportions.
• Manipulate the equation to isolate the unknown variable.

Environmental Conservation

Environmental conservation refers to the practice of using natural resources sustainably to protect the environment. In this context, recycling newspapers to save trees is a clear example.
Trees, such as loblolly pines, are critical for ecosystems as they provide habitat, produce oxygen, and store carbon. By recycling, we reduce the need to cut down these trees, thus contributing to conservation efforts.
Understanding the impact of recycling on tree conservation can motivate individuals and societies to engage in more sustainable practices.

• Recycling reduces waste and saves natural resources.
• Every 3.5-foot recycled newspaper stack can save a 20-foot tree.
• Promotes biodiversity by preserving vital habitats for wildlife.

Through this exercise, not only do we solve a mathematical problem, but we also engage with an essential ecological responsibility.