Answer
$c=\frac{2}{3}$
So, the ball is hollow.
Work Step by Step
We know that the translational kinetic energy is given as
$K_t=\frac{1}{2}mv^2$
The rotational kinetic energy is given as
$K_r=\frac{1}{2}I\omega^2=\frac{1}{2}(cmr^2)(\frac{v}{r})^2=\frac{1}{2}cmv^2$
Now the total kinetic energy is:
$K_{total}=K_t+K_r$
$K_{total}=\frac{1}{2}mv^2+\frac{1}{2}cmv^2=\frac{1}{2}(c+1)mv^2$
Thus, the required ratio is
$\frac{K_r}{K_{total}}=\frac{40}{100}=0.4=\frac{\frac{1}{2}cmv^2}{\frac{1}{2}(c+1)mv^2}$
This simplifies to:
$c=\frac{0.4}{0.6}$
$c=\frac{2}{3}$
So, the ball is hollow.
