Show that β ≈ 3α, by calculating the change in volume ΔV of a cube with sides of length L.
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Show that $\beta \approx 3 \alpha,$ by calculating the change in volume $\Delta V$ of a cube with sides of length $L$
Volume The change in the volume $V=x^{3}$ of a cube when the edge lengths change from $a$ to $a+d x$
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The volume of a solid cube with side $s_{0}$ at temperature $T_{0}$ is $V_{0}=s_{0}^{3} .$ Show that if $\Delta s < s_{0},$ the change in volume $\Delta V$ due to a change in temperature $\Delta T$ is given by $$ \frac{\Delta V}{V_{0}}=3 \alpha \Delta T $$ and therefore that $\beta=3 \alpha .$ (Although we derive this relation for a cube, it applies to a solid of any shape.)
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