Answer
\[\kappa = \frac{{2\sqrt 5 }}{{{{\left( {20{{\sin }^2}t + {{\cos }^2}t} \right)}^{3/2}}}}\]
Work Step by Step
\[\begin{gathered}
{\mathbf{r}}\left( t \right) = \left\langle {4\cos t,\sin t,2\cos t} \right\rangle \hfill \\
{\text{Calculate }}{\mathbf{v}}\left( t \right){\text{ and }}\left| {{\mathbf{v}}\left( t \right)} \right| \hfill \\
{\mathbf{v}}\left( t \right) = {\mathbf{r}}'\left( t \right) \hfill \\
{\mathbf{v}}\left( t \right) = \frac{d}{{dt}}\left\langle {4\cos t,\sin t,2\cos t} \right\rangle \hfill \\
{\mathbf{v}}\left( t \right) = \left\langle { - 4\sin t,\cos t, - 2\sin t} \right\rangle \hfill \\
\left| {{\mathbf{v}}\left( t \right)} \right| = \sqrt {{{\left( { - 4\sin t} \right)}^2} + {{\left( {\cos t} \right)}^2} + {{\left( { - 2\sin t} \right)}^2}} \hfill \\
\left| {{\mathbf{v}}\left( t \right)} \right| = \sqrt {16{{\sin }^2}t + {{\cos }^2}t + 4{{\sin }^2}t} \hfill \\
\left| {{\mathbf{v}}\left( t \right)} \right| = \sqrt {20{{\sin }^2}t + {{\cos }^2}t} \hfill \\
\hfill \\
{\text{Calculate }}{\mathbf{a}}\left( t \right) \hfill \\
{\mathbf{a}}\left( t \right) = {\mathbf{v}}'\left( t \right) \hfill \\
{\mathbf{a}}\left( t \right) = \frac{d}{{dt}}\left\langle { - 4\sin t,\cos t, - 2\sin t} \right\rangle \hfill \\
{\mathbf{a}}\left( t \right) = \left\langle { - 4\cos t, - \sin t, - 2\cos t} \right\rangle \hfill \\
\hfill \\
{\text{Calculate }}\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right| \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
{ - 4\sin t}&{\cos t}&{ - 2\sin t} \\
{ - 4\cos t}&{ - \sin t}&{ - 2\cos t}
\end{array}} \right| \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = \left| {\begin{array}{*{20}{c}}
{\cos t}&{ - 2\sin t} \\
{ - \sin t}&{ - 2\cos t}
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
{ - 4\sin t}&{ - 2\sin t} \\
{ - 4\cos t}&{ - 2\cos t}
\end{array}} \right|{\mathbf{j}} \hfill \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left| {\begin{array}{*{20}{c}}
{ - 4\sin t}&{\cos t} \\
{ - 4\cos t}&{ - \sin t}
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = \left( { - 2{{\cos }^2}t - 2{{\sin }^2}t} \right){\mathbf{i}} - \left( {8\sin t\cos t - 8\sin t\cos t} \right){\mathbf{j}} \hfill \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {4{{\sin }^2}t + 4{{\cos }^2}t} \right){\mathbf{k}} \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = - 2{\mathbf{i}} + 0{\mathbf{j}} + 4{\mathbf{k}} \hfill \\
\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right| = \sqrt {4 + 16} \hfill \\
\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right| = \sqrt {20} \hfill \\
\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right| = 2\sqrt 5 \hfill \\
\hfill \\
{\text{Use the alternative curvature formula}} \hfill \\
\kappa = \frac{{\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right|}}{{{{\left| {{\mathbf{v}}\left( t \right)} \right|}^3}}} \hfill \\
\kappa = \frac{{2\sqrt 5 }}{{{{\left( {\sqrt {20{{\sin }^2}t + {{\cos }^2}t} } \right)}^3}}} \hfill \\
\kappa = \frac{{2\sqrt 5 }}{{{{\left( {20{{\sin }^2}t + {{\cos }^2}t} \right)}^{3/2}}}} \hfill \\
\end{gathered} \]
