Answer
$\dfrac{2 \sqrt {74}}{27}$
Work Step by Step
The curvature $\kappa$ for a plane curve system is:
$\kappa (t)= \dfrac{||r'(t) \times r''(t)||}{||r'(t)||^3}$
We have:
$r'(t) =\lt -t^{-2}, -2t^{-3}, 2t \gt$ and $r''(t) =\lt 2t^{-3}, 6t^{-4}, 2 \gt$
Thus,
$\kappa(t) = \dfrac{||16t^{-3}i+6t^{-2}j-2t^{-6}k ||}{||\lt -t^{-2}, -2t^{-3}, 2t \gt||^3} \\=\dfrac{\sqrt {256t^{-6}+36t^{-4}+4t^{-12}}}{[(-t^{-4})^2+4t^{-6}+4t^2]^{3/2}}$
Now, we will compute $\kappa (t)$ at $t=1$
Therefore,
$\kappa(t)= \dfrac{\sqrt {296}}{(9)^{3/2}}=\dfrac{2 \sqrt {74}}{27}$
