To analyze the thermal expansion of a metal rod, we usually refer to the formula for linear expansion, which is:
\[ \Delta L = \alpha L_0 \Delta T \]
where:
- \(\Delta L\) is the change in length (final length \(L_2\) minus initial length \(L_1\))
- \(L_0\) is the initial length of the rod (assumed as \(L_1\))
- \(\alpha\) (or \(a\)) is the linear expansivity of the material
- \(\Delta T\) is the change in temperature (final temperature \(t_2\) minus initial temperature \(t_1\))
Given these concepts, you can express the change in length due to temperature changes as:
\[ L_2 - L_1 = \alpha L_1 (t_2 - t_1) \]
If we denote the temperature change as \(t\) (where \(t = t_2 - t_1\)), then we can rewrite:
\[ L_2 - L_1 = \alpha L_1 t \]
From your options, the shaded part can represent the relevant length changes or related quantities based on the linear expansion formula.
The correct interpretation based on the choices provided appears to be:
- If considering \(L_1\) as the initial length and using the linear expansivity \(a\) and temperature change \(t\), the expression that aligns with the expanded length relative to the initial length would be:
\[ L_2 - L_1 = a L_1 t \]
Thus, considering that the shaded area could represent the effective change in length due to thermal expansion, we can conclude:
The correct choice is d. \(a(L_2 - L_1)t\), assuming \(L_2 - L_1\) represents the changed segment due to this thermal expansion. Be sure to clarify the terms and context of variables since the symbolization may vary.
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