Answer
$\beta = 0.07307$
Work Step by Step
We can find $\gamma$:
$K = mc^2~(\gamma-1) = 10.00~MeV$
$\gamma-1 = \frac{10.00~MeV}{mc^2}$
$\gamma = 1+\frac{10.00~MeV}{E_0}$
$\gamma = 1+\frac{10.00~MeV}{3727.40~MeV}$
$\gamma = 1.00268$
We can find $\beta$:
$\gamma = \frac{1}{\sqrt{1-\beta^2}}$
$1-\beta^2 = \frac{1}{\gamma^2}$
$\beta^2 = 1-\frac{1}{\gamma^2}$
$\beta = \sqrt{1-\frac{1}{\gamma^2}}$
$\beta = \sqrt{1-\frac{1}{(1.00268)^2}}$
$\beta = 0.07307$
