An electron gun sends out electrons that have been sped up through 180 V from almost rest…

Step 1. The speed. We want $v$, so $v = \sqrt{\dfrac{2eV}{m}} = \sqrt{\dfrac{2 \times 1.6 \times 10^{-19}\ \text{C} \times 180\ \text{V}}{9.1 \times 10^{-31}\ \text{kg}}} = 7.96 \times 10^{6}\ \text{m/s}$.

Step 2. The radius. The top of $r = \dfrac{mv}{eB}$ is $mv = 9.1 \times 10^{-31}\ \text{kg} \times 7.96 \times 10^{6}\ \text{m/s} = 7.24 \times 10^{-24}\ \text{kg m/s}$. The bottom is $eB = 1.6 \times 10^{-19}\ \text{C} \times 2.4 \times 10^{-3}\ \text{T} = 3.84 \times 10^{-22}\ \text{C T}$. So $r = \dfrac{7.24 \times 10^{-24}\ \text{kg m/s}}{3.84 \times 10^{-22}\ \text{C T}} = 1.89 \times 10^{-2}\ \text{m}$, that is $1.89$ cm.

Step 3. The period. $T = \dfrac{2\pi m}{eB} = \dfrac{2\pi \times 9.1 \times 10^{-31}\ \text{kg}}{3.84 \times 10^{-22}\ \text{C T}} = 1.49 \times 10^{-8}\ \text{s}$, that is $14.9$ ns.