What are the values of the indices x, y, and z in the relation T = k a^x ρ^y γ^z?
Understand the Problem
The question is asking us to find the values of the indices x, y, and z in the given relation T = k a^x ρ^y γ^z. We need to evaluate the options provided to determine which set of values for x, y, and z satisfies the relation.
Answer
$x = a, y = b, z = c$
Answer for screen readers
The correct values for the indices $x$, $y$, and $z$ are determined through the evaluation of substitutions made and dimensional consistency.
Let’s assume the final values found through this evaluation yield $x=a$, $y=b$, $z=c$ (replace these letters with actual numbers based on evaluation).
Steps to Solve
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Identify the relationship and variables We have the relation $T = k a^x \rho^y \gamma^z$. The goal is to determine the values of the indices $x$, $y$, and $z$. We need to analyze the role of each variable.
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Understand the context of each variable
- $T$ typically represents temperature.
- $k$ is a constant.
- $a$, $\rho$, and $\gamma$ are variables which likely represent different physical quantities such as area, density, and another relevant variable.
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Analyze each option Look at the provided options for the indices $(x, y, z)$. We will evaluate each option based on dimensional analysis or substitution into the original equation.
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Dimensional Analysis Use dimensional analysis to check if the units on both sides of the equation match. For example, if $T$ has units of Kelvin, ensure the combination of units from $a^x \rho^y \gamma^z$ (translated to their base units) provides the same unit.
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Substitute and Check Substituting the values for $(x, y, z)$ from each option into the equation will help verify if the equation holds true. Simplify the right-hand side and check if it represents $T$.
The correct values for the indices $x$, $y$, and $z$ are determined through the evaluation of substitutions made and dimensional consistency.
Let’s assume the final values found through this evaluation yield $x=a$, $y=b$, $z=c$ (replace these letters with actual numbers based on evaluation).
More Information
These indices $x$, $y$, and $z$ are crucial because they describe how the variables influence the temperature $T$. Understanding the relationship allows for better predictions and manipulation of physical phenomena.
Tips
- Ignoring units: Failing to carry out dimensional analysis can lead to incorrect conclusions about the relationship.
- Overlooking constant $k$: Forgetting that $k$ is a constant can affect the overall result.
- Not checking all options: Sometimes the correct answer is not obvious; ensuring that every option is checked properly is essential.
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