A current-carrying solenoid has a paramagnetic core. At a...

The percentage increase in the magnetic field due to the insertion of a core is given by:
$\Delta B (\%) = \chi \times 100\%$
where $\chi$ is the magnetic susceptibility of the core material.

For a paramagnetic material, Curie's Law states that the magnetic susceptibility is inversely proportional to the absolute temperature $T$ :
$\chi \propto \dfrac{1}{T}$

Therefore, the percentage increase in the magnetic field is also inversely proportional to the temperature:
$\dfrac{\Delta B_2 (\%)}{\Delta B_1 (\%)} = \dfrac{T_1}{T_2}$

Given $T_1 = 300\text{ K}$, $\Delta B_1 (\%) = 0.06\%$, and $T_2 = 450\text{ K}$ :
$\Delta B_2 (\%) = 0.06\% \times \dfrac{300}{450}$
$\Delta B_2 (\%) = 0.06\% \times \dfrac{2}{3} = 0.04\%$

Answer: $0.04\%$