A steel bolt and a brass bolt are placed side-by-side with a 5.0 μm gap between them at 28.70°C. If the temperature increases, how much must the temperature increase for the bolts to touch? The coeffi

Initial gap, $\text{gap} = 5.0 \, \mu\text{m} = 5.0 \times 10^{-6} \, \text{m}$
Initial temperature, $T_0 = 28.70^\circ\text{C}$
Coefficient of linear expansion for steel, $\alpha_{\text{steel}} = 11 \times 10^{-6} \, (^\circ\text{C})^{-1}$
Coefficient of linear expansion for brass, $\alpha_{\text{brass}} = 19 \times 10^{-6} \, (^\circ\text{C})^{-1}$

The change in length for each bolt is given by the linear thermal expansion formula:

For the steel bolt:

For the brass bolt:

where $\Delta T = T_f - T_0$ is the change in temperature.

The bolts will touch when the sum of their expansions equals the initial gap:

Substituting the expansion formulas:

Factoring out $\Delta T$:

Solving for $\Delta T$:

The final temperature $T_f$ would be:

a) At what temperature will the bolts touch?
As shown in Step 3, to calculate a numerical value for $\Delta T$ and thus $T_f$, the initial lengths of the steel bolt ($L_{0, \text{steel}}$) and the brass bolt ($L_{0, \text{brass}}$) are required. Since these values are not provided in the problem statement, a numerical answer cannot be determined.

b) determine respectively, the change in the length of the steel and the brass at that temperature.
Similarly, without the initial lengths ($L_{0, \text{steel}}$ and $L_{0, \text{brass}}$) and the calculated $\Delta T$, the individual changes in length ($\Delta L_{\text{steel}}$ and $\Delta L_{\text{brass}}$) cannot be determined numerically.

Please provide the initial lengths of the bolts if you would like a numerical solution.