Answer
360.5$^{\circ}$C
or 733.7 K
Work Step by Step
Given/Needed from charts:
Rod is steel, ring is brass. I will represent them with s for steel and b for brass
Ts = 25.00$^{\circ}$C
Ls = 3.000 cm
Tb = 25.00$^{\circ}$C
Lb = 2.992 cm
$\alpha$b = 19 * $10^{-6}$ $\frac{1}{{\circ}C}$
$\alpha$s = 11 * $10^{-6}$ $\frac{1}{{\circ}C}$
We will need this equation:
$\Delta$L = Lf - Li = L$\alpha$$\Delta$T
Also stated as
Lf = L$\alpha$$\Delta$T + Li
For the ring to fit, the Tf of the brass ring must equal the Tf of the steel rod. Thus,
Ls$\alpha$s$\Delta$T + Lis = Lb$\alpha$b$\Delta$T + Lib
Plug in:
(3)(11 * $10^{-6}$)$\Delta$T + 3 = (2.992) (19 * $10^{-6}$)$\Delta$T+2.992
Simplifying:
(33 * $10^{-6}$)$\Delta$T + 0.008 = (56.848 * $10^{-6}$)$\Delta$T
Bring $\Delta$T to one side and factor out the coefficients:
$\Delta$T*(56.848 * $10^{-6}$ - 33 * $10^{-6}$) = 0.008
Thus:
$\Delta$T = $\frac{0.008}{(56.848 * 10^{-6} -33 * 10^{-6})}$ = 335.5$^{\circ}$C
This is a change in temperature, so you can represent it as either Kelvin or Celsius. Since they are asking for the final temperature, not the change in temperature:
Ts + $\Delta$T = 360.5$^{\circ}$C
Or, converting to SI units:
360.5$^{\circ}$C + 373.2$^{\circ}$C = 733.7 K
