Determine the value of \$\text{C}\cdot\text{L}\$ in a LC circuit
An LC resonant circuit has a resonant frequency of \$\dfrac{1}{2\pi\sqrt{LC}}\$.

So, the period of this is \$2\pi\sqrt{LC}\$

If you want half that time then t = \$\pi\sqrt{LC}\$

If you want this equating to 10 us then \$LC = \dfrac{10us^2}{\pi^2}\$

10us is \$10^{-5}\$ and not \$10^{-7}\$.

That's the only mistake I can see.