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(C) Assuming Hooke's law is valid,the tension $T$ is proportional to the extension $\Delta \ell = \ell - \ell_{0}$,where $\ell_{0}$ is the original le

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$\frac{T_{2}\ell_{1}-T_{1}\ell_{2}}{T_{2}-T_{1}}$

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(C) Assuming Hooke's law is valid,the tension $T$ is proportional to the extension $\Delta \ell = \ell - \ell_{0}$,where $\ell_{0}$ is the original length.$T = k(\ell - \ell_{0})$For the two given states:$T_{1} = k(\ell_{1} - \ell_{0})$  --- $(1)$$T_{2} = k(\ell_{2} - \ell_{0})$  --- $(2)$Dividing equation $(1)$ by $(2)$:$\frac{T_{1}}{T_{2}} = \frac{\ell_{1} - \ell_{0}}{\ell_{2} - \ell_{0}}$Cross-multiplying:$T_{1}(\ell_{2} - \ell_{0}) = T_{2}(\ell_{1} - \ell_{0})$$T_{1}\ell_{2} - T_{1}\ell_{0} = T_{2}\ell_{1} - T_{2}\ell_{0}$Rearranging to solve for $\ell_{0}$:$T_{2}\ell_{0} - T_{1}\ell_{0} = T_{2}\ell_{1} - T_{1}\ell_{2}$$\ell_{0}(T_{2} - T_{1}) = T_{2}\ell_{1} - T_{1}\ell_{2}$$\ell_{0} = \frac{T_{2}\ell_{1} - T_{1}\ell_{2}}{T_{2} - T_{1}}$

The length of a metallic wire is $\ell_{1}$ when the tension in it is $T_{1}$. It is $\ell_{2}$ when the tension is $T_{2}$. The original length of the wire will be ...... .

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