At $20\ ^\circ\text{C}$, a round hole in a steel plate is $2.000\ \text{cm}$ across, and a brass ball is $2.004\ \text{cm}$ across, as shown in the figure. For steel $\alpha = 1.2 \times 10^{-5}\ \text{K}^{-1}$, and for brass $\alpha = 1.9 \times 10^{-5}\ \text{K}^{-1}$. The ball is kept at $20\ ^\circ\text{C}$. To what temperature must the plate be heated so that the ball can just pass through the hole?

Fig. 1: A steel plate with a round hole 2.000 cm across, and beside it a brass ball 2.004 cm across, both at 20 degrees Celsius.