- Define the relationship between cans recycled (\(C\)) and energy saved (\(W\)) as a proportional function: \(W = k \times C\).
- Calculate the constant of proportionality (\(k\)) using the given data: \(k = \frac{2 \text{ weeks}}{100 \text{ cans}} = 0.02 \text{ weeks/can}\).
- Apply the constant to find the savings for 200 cans: \(W = 0.02 \times 200 = 4 \text{ weeks}\).
- The total energy saved by recycling 200 cans is \(\boxed{4}\).
Explanation
- Problem Analysis and Modeling
The objective is to find the energy saved in weeks by recycling 200 aluminum cans, given that recycling 100 aluminum cans saves enough energy for 2 weeks. We can identify the number of cans recycled as \(C\) and the energy saved in weeks as \(W\). Since the problem implies a constant rate of energy savings per can, we can model this relationship as a direct proportionality, \(W = k \times C\), where \(k\) is the constant of proportionality (the number of weeks of energy saved per can).
- Determining the Proportionality Constant
Given that 100 cans yield 2 weeks of energy savings, we can find the constant \(k\) by substituting these values into our equation:
\[2 = k \times 100\]
Solving for \(k\):
\[k = \frac{2}{100} = 0.02 \text{ weeks/can.}\]
This means that every single can recycled contributes to saving 0.02 weeks of energy.
- Calculating the Result
Now that we have established the constant \(k = 0.02\), we can determine the energy saved for 200 cans by substituting \(C = 200\) into our proportional equation:
\[W = 0.02 \times 200\]
Calculating this:
\[W = 4\]
Therefore, recycling 200 cans saves energy for 4 weeks. The final answer is \(\boxed{4}\).
Examples
Understanding proportional relationships is crucial for scaling up resources in many real-life projects. For instance, if you are planning a community garden and know that one packet of seeds produces 10 pounds of tomatoes, you can use the same proportionality logic to predict that 5 packets of seeds will produce \(10 \times 5 = 50\) pounds of tomatoes. This skill allows you to estimate requirements for larger goals efficiently and accurately, just as we did to calculate the energy savings for a larger recycling effort.