Answer
$$\kappa \left( t \right) = \frac{4}{{17}}$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = 4\cos t{\bf{i}} + 4\sin t{\bf{j}} + t{\bf{k}} \cr
& {\text{Calculate the derivatives }}{\bf{r}}'\left( t \right){\text{ and }}{\bf{r}}{\text{''}}\left( t \right) \cr
& {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {4\cos t} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {4\sin t} \right]{\bf{j}} + \frac{d}{{dt}}\left[ t \right]{\bf{k}} \cr
& {\bf{r}}'\left( t \right) = - 4\sin t{\bf{i}} + 4\cos t{\bf{j}} + \left( 1 \right){\bf{k}} \cr
& and \cr
& {\bf{r}}''\left( t \right) = \frac{d}{{dt}}\left[ { - 4\sin t} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {4\cos t} \right]{\bf{j}} + \frac{d}{{dt}}\left[ 1 \right]{\bf{k}} \cr
& {\bf{r}}''\left( t \right) = - 4\cos t{\bf{i}} - 4\sin t{\bf{j}} \cr
& \cr
& {\text{Calculate the cross product }}{\bf{r}}'\left( t \right) \times {\bf{r}}''\left( t \right) \cr} $$
\[\begin{gathered}
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
{ - 4\sin t}&{4\cos t}&1 \\
{ - 4\cos t}&{ - 4\sin t}&0
\end{array}} \right| \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
{4\cos t}&1 \\
{ - 4\sin t}&0
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
{ - 4\sin t}&1 \\
{ - 4\cos t}&0
\end{array}} \right|{\mathbf{j}} + \left| {\begin{array}{*{20}{c}}
{ - 4\sin t}&{4\cos t} \\
{ - 4\cos t}&{ - 4\sin t}
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left( {0 + 4\sin t} \right){\mathbf{i}} - \left( {0 + 4\cos t} \right){\mathbf{j}} + \left( {16{{\sin }^2}t + 16{{\cos }^2}t} \right){\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = 4\sin t{\mathbf{i}} - 4\cos t{\mathbf{j}} + 16{\mathbf{k}} \hfill \\
\end{gathered} \]
$$\eqalign{
& {\text{Use the formula }}\left( 3 \right)\,\,\,\kappa \left( t \right) = \frac{{\left\| {{\bf{r}}'\left( t \right) \times {\bf{r}}''\left( t \right)} \right\|}}{{{{\left\| {{\bf{r}}'\left( t \right)} \right\|}^3}}}.{\text{ Thus}}{\text{,}} \cr
& \kappa \left( t \right) = \frac{{\left\| {4\sin t{\bf{i}} - 4\cos t{\bf{j}} + 16{\bf{k}}} \right\|}}{{{{\left\| { - 4\sin t{\bf{i}} + 4\cos t{\bf{j}} + \left( 1 \right){\bf{k}}} \right\|}^3}}} \cr
& \kappa \left( t \right) = \frac{{\sqrt {16{{\sin }^2}t + 16{{\cos }^2}t + {{16}^2}} }}{{{{\left( {\sqrt {16{{\sin }^2}t + 16{{\cos }^2}t + 1} } \right)}^3}}} \cr
& \kappa \left( t \right) = \frac{{\sqrt {16 + {{16}^2}} }}{{{{\left( {\sqrt {16 + 1} } \right)}^3}}} \cr
& \kappa \left( t \right) = \frac{{\sqrt {272} }}{{{{\left( {\sqrt {17} } \right)}^{3/2}}}} \cr
& \kappa \left( t \right) = \frac{4}{{17}} \cr} $$
