Analyze the Worked Examples: Identifying Input, Output, and Constant of Proportionality
1. Understanding the Scenario
In this scenario, Kim discovers that for every 3 minutes she spends in the shower, she uses 25 gallons of water. The problem states that this rate is constant, which means that there is a steady relationship between time spent in the shower and water usage.
To solve the problem, we need to identify three things:
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Input: This is the value we will start with or the independent variable we can control. In this case, it’s the time Kim spends in the shower.
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Output: This is the dependent variable that results from our input. Here, it’s the amount of water Kim uses.
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Constant of Proportionality: Known as the rate, this constant is derived by dividing the output by the input. It’s the amount of water used per minute in the shower.
2. Identifying Input, Output, and Constant of Proportionality
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Input (Independent Variable): The input for this specific problem is the time Kim spends in the shower. Based on the information provided, the input is measured in minutes. Therefore, if Kim wants to determine the amount of water used, she will input the time in minutes into her equation.
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Output (Dependent Variable): The output is the total amount of water used, which will change based on how many minutes Kim spends in the shower. This is measured in gallons.
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Constant of Proportionality: The constant of proportionality represents the rate of water usage per minute. To find this, we divide the gallons of water used by the time in minutes. Given that 3 minutes result in the usage of 25 gallons, the constant ( k ) can be calculated as:
k = \frac{\text{Total Water Used in Gallons}}{\text{Total Time in Minutes}} = \frac{25 \text{ gallons}}{3 \text{ minutes}} \approx 8.33 \text{ gallons per minute}
3. Applying the Understanding to Problem Solving
Once we’ve identified these values, Kim can use this information to predict water usage for any given time in the shower. This helps in forming a proportional relationship written as:
\text{Water Usage (gallons)} = k \times \text{Time Spent (minutes)}
Plugging in the constant value:
\text{Water Usage (gallons)} = 8.33 \times \text{Time Spent (minutes)}
This formula can be used to calculate the number of gallons used for any period Kim decides to spend in the shower.
4. Example Scenarios
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Scenario 1: Suppose Kim showers for 6 minutes. Using the formula:
\text{Water Usage} = 8.33 \text{ gallons/minute} \times 6 \text{ minutes} = 49.98 \text{ gallons}Therefore, Kim would use approximately 50 gallons of water for a 6-minute shower.
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Scenario 2: If Kim showers for only 1.5 minutes, the water usage would be:
\text{Water Usage} = 8.33 \text{ gallons/minute} \times 1.5 \text{ minutes} = 12.495 \text{ gallons}Hence, for a 1.5-minute shower, the usage is approximately 12.5 gallons of water.
5. Understanding Proportionality in Real Life
Understanding proportionality helps in managing resources more effectively. In real life scenarios, this knowledge allows Kim to:
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Conserve Water: By understanding her rate of consumption, Kim can better plan and reduce water usage.
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Budget Planning: Knowing the water usage and its cost per gallon can help Kim manage her finances efficiently.
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Time Management: By controlling the shower time, Kim can balance her daily schedule considering her busier days.
Summary
Through this analysis, we’ve discovered how input, output, and a constant rate influence proportional relationships. By mastering these concepts, one can predict changes in one variable based on adjustments in another, understanding real-life applications like water conservation or financial budgeting. By applying these calculations, Kim and others can make informed decisions that potentially lead to water saving practices.
Please let me know if you have any more questions or need further clarification, @Rain_Bulade.