Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \langle \sin 2 t , 3 t , \cos 2 t \rangle \text { when } t = \frac { \pi } { 2 }$
A) $\frac { 3 } { 13 }$

B) $\frac { 4 } { 13 }$
C) $\frac { 6 } { 13 }$
D) $\frac { 8 } { 13 }$
E) $\frac { 1 } { 3 }$
F) $\frac { 4 } { 9 }$

G) $\frac { 2 } { 3 }$

H) $\frac { 8 } { 9 }$

Question

Find the curvature $K$ of the curve $\mathbf { r } ( t ) = \langle \sin 2 t , 3 t , \cos 2 t \rangle \text { when } t = \frac { \pi } { 2 }$ 
A) $\frac { 3 } { 13 }$ 
 
B) $\frac { 4 } { 13 }$
C) $\frac { 6 } { 13 }$  
D) $\frac { 8 } { 13 }$ 
E) $\frac { 1 } { 3 }$
F) $\frac { 4 } { 9 }$

G) $\frac { 2 } { 3 }$

H) $\frac { 8 } { 9 }$

Quizplus · Accepted Answer

The curvature $K$ of a curve $\mathbf{r}(t)$ is given by $K = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}$. First, find the first and second derivatives of $\mathbf{r}(t)$, then evaluate these at $t = \frac{\pi}{2}$, and finally, compute the curvature using the formula. The correct calculation leads to $\frac{4}{13}$ for $t = \frac{\pi}{2}$.

Find the Curvature KKK Of the Curve r(t)=⟨sin⁡2t,3t,cos⁡2t⟩ when t=π2\mathbf { r } ( t ) = \langle \sin 2 t , 3 t , \cos 2 t \rangle \text { when } t = \frac { \pi } { 2 }r(t)=⟨sin2t,3t,cos2t⟩ when t=2π​

Find the Curvature $K$ Of the Curve $\mathbf { r } ( t ) = \langle \sin 2 t , 3 t , \cos 2 t \rangle \text { when } t = \frac { \pi } { 2 }$