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Answer: The coefficients of areal and volumetric expansion are related to the coefficient of linear expansion for a material. Here's how they compare for metals A and B given that the coefficient of linear expansion of metal A (\(\alpha_A\)) is three times that of metal B (\(\alpha_B\)):
### Areal Expansion
The coefficient of areal expansion (\(\beta\)) is related to the coefficient of linear expansion (\(\alpha\)) by the formula:
\[ \beta = 2\alpha \]
Given that:
\[ \alpha_A = 3\alpha_B \]
The areal expansion coefficients for metals A and B are:
\[ \beta_A = 2\alpha_A \]
\[ \beta_B = 2\alpha_B \]
Substituting \(\alpha_A = 3\alpha_B\):
\[ \beta_A = 2 \times 3\alpha_B = 6\alpha_B \]
So, the areal expansion coefficient of metal A (\(\beta_A\)) is:
\[ \beta_A = 6\alpha_B \]
This means that:
\[ \beta_A = 3 \beta_B \]
Thus, the coefficient of areal expansion of metal A is three times that of metal B.
### Volumetric Expansion
The coefficient of volumetric expansion (\(\gamma\)) is related to the coefficient of linear expansion (\(\alpha\)) by the formula:
\[ \gamma = 3\alpha \]
Given that:
\[ \alpha_A = 3\alpha_B \]
The volumetric expansion coefficients for metals A and B are:
\[ \gamma_A = 3\alpha_A \]
\[ \gamma_B = 3\alpha_B \]
Substituting \(\alpha_A = 3\alpha_B\):
\[ \gamma_A = 3 \times 3\alpha_B = 9\alpha_B \]
So, the volumetric expansion coefficient of metal A (\(\gamma_A\)) is:
\[ \gamma_A = 9\alpha_B \]
This means that:
\[ \gamma_A = 3 \gamma_B \]
Thus, the coefficient of volumetric expansion of metal A is three times that of metal B.
### Summary
- The coefficient of areal expansion (\(\beta\)) of metal A is three times that of metal B.
- The coefficient of volumetric expansion (\(\gamma\)) of metal A is three times that of metal B.
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