Suppose That M Varies Directly As Z, And M 120 When Z 15. Write An Equation That Expresses This Variation.
Understanding how variables relate to each other is a fundamental aspect of algebra and mathematical modeling. When a variable is said to vary directly as another, it indicates a linear relationship between them, meaning the ratio of the two variables remains constant. In this context, the problem states that M varies directly as Z, and provides a specific data point: M = 120 when Z = 15. The goal is to derive an equation that models this variation accurately.
This article will guide you through the process of formulating the direct variation equation, understanding the concept of direct variation, applying the given data, and interpreting the resulting formula. We will explore the mathematics behind direct variation, methods to determine the constant of variation, and practical examples to solidify your understanding.
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Understanding Direct Variation
What Is Direct Variation?
Direct variation describes a relationship between two variables where one variable increases or decreases proportionally with the other. Mathematically, this relationship can be expressed as:
\[ M = kZ \]
where:
- M is the dependent variable,
- Z is the independent variable,
- k is the constant of variation (also called the constant of proportionality).
This means that if Z doubles, then M also doubles; if Z triples, M triples, and so on.
Characteristics of Direct Variation
Variables that vary directly exhibit certain key features:
- Linear Relationship: The graph of M versus Z is a straight line passing through the origin (0,0).
- Constant Ratio: The ratio \( \frac{M}{Z} \) remains constant for all values of Z.
- Proportionality: The change in M is directly proportional to the change in Z.
Understanding these features is essential for modeling real-world situations, such as physics problems involving speed, proportional costs, or geometric relationships.
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Formulating the Equation of Direct Variation
Step 1: Recognize the Relationship
Given that M varies directly as Z, the general form is:
\[ M = kZ \]
where k is an unknown constant that needs to be determined based on the data.
Step 2: Use the Given Data to Find the Constant \(k\)
The problem provides a specific data point:
- When Z = 15, M = 120.
Substitute these values into the general formula:
\[ 120 = k \times 15 \]
To solve for \(k\):
\[ k = \frac{120}{15} \]
\[ k = 8 \]
Step 3: Write the Specific Equation
Now that you have determined the value of \(k\), substitute it back into the general form:
\[ M = 8Z \]
This is the equation that models the direct variation between M and Z based on the given data.
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Interpreting the Equation \( M = 8Z \)
Meaning of the Constant of Variation \(k=8\)
In this context, \(k=8\) signifies that for every unit increase in Z, M increases by 8 units. The constant of variation indicates the rate of change or the proportionality factor between the variables.
Graphical Representation
- The graph of \( M = 8Z \) is a straight line passing through the origin.
- The slope of the line is 8, indicating the rate at which M increases with Z.
- For Z values of 0, M will also be 0, consistent with the property of direct variation.
Practical Examples
Suppose Z represents the number of hours worked, and M represents earnings, with a rate of $8 per hour. If Z is 15 hours, earnings M are:
\[ M = 8 \times 15 = 120 \]
This example illustrates how the model can predict or describe real-world scenarios.
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Extensions and Applications of the Concept
1. Verifying the Model with Additional Data
To ensure the accuracy of the model, you can verify it with other data points if available. For example, if Z = 10, then:
\[ M = 8 \times 10 = 80 \]
If actual M is close to 80 when Z is 10, this confirms the model's validity.
2. Using the Equation in Problem Solving
The derived equation can be used to:
- Predict M for other Z values.
- Find Z given M.
- Analyze how changes in Z affect M.
For instance, if M is 200, then:
\[ 200 = 8Z \]
\[ Z = \frac{200}{8} = 25 \]
Thus, Z must be 25 to achieve M of 200.
3. Real-World Applications of Direct Variation
Direct variation models are prevalent in various fields:
- Physics: Speed proportional to distance over time.
- Economics: Cost proportional to quantity.
- Engineering: Resistance proportional to material length.
- Biology: Rate of enzyme activity proportional to substrate concentration.
Understanding the mathematical foundation allows for accurate modeling and decision-making across disciplines.
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Common Mistakes and Tips
1. Confusing Direct and Inverse Variation
- Direct Variation: \( M = kZ \), where M increases with Z.
- Inverse Variation: \( M = \frac{k}{Z} \), where M decreases as Z increases.
2. Miscalculating the Constant \(k\)
Always plug in the known data correctly and perform the division carefully to find the constant of variation.
3. Assuming the Line Passes Through the Origin When Not Given Data
In pure direct variation, the line always passes through (0,0). However, if the data point does not satisfy \( M=0 \) when \( Z=0 \), then the variation might not be purely direct, or additional factors may be involved.
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Summary and Conclusion
In summary, when a variable M varies directly as Z, the relationship can be modeled with the equation:
\[ M = kZ \]
Given the specific data point \( M=120 \) when \( Z=15 \), we determine:
\[ k = \frac{120}{15} = 8 \]
and thus, the variation is accurately modeled by:
\[ \boxed{ M = 8Z } \]
This simple yet powerful formula allows for easy calculation of M for any Z, facilitates understanding of proportional relationships, and provides a foundation for solving more complex problems involving direct variation.
By mastering the process of deriving such equations, you enhance your algebraic problem-solving skills and your ability to interpret real-world relationships through mathematics.
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Final Note: Always verify the assumptions of the model with the context of the problem, and remember that real-world data may sometimes deviate from ideal models. Use the derived equations as tools for approximation and understanding, and refine them as needed based on additional information.
