91Ó°ÊÓ

\(39-44=\) Find the derivative of the vector function. $$ \mathbf{r}(t)=\left\langle t \sin t, t^{2}, t \cos 2 t\right\rangle $$

A sled is pulled along a level path through snow by a rope. A 30 -lb force acting at an angle of \(40^{\circ}\) above the horizontal moves the sled 80 ft. Find the work done by the force.

(a) Find parametric equations for the line of intersection of the planes \(x+y+z=1\) and \(x+2 y+2 z=1\) (b) Find the angle between these planes.

Find the vectors \(\mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given point. $$\mathbf{r}(t)=\left\langle t^{2}, \frac{2}{3} t^{3}, t\right\rangle, \quad\left(1, \frac{2}{3}, 1\right)$$

Find the vectors \(\mathbf{T}, \mathbf{N},\) and \(\mathbf{B}\) at the given point. $$\mathbf{r}(t)=\langle\cos t, \sin t, \ln \cos t\rangle, \quad(1,0,0)$$

A boat sails south with the help of a wind blowing in the direction \(S36^{\circ} \mathrm{E}\) with magnitude 400 \(\mathrm{lb}\) . Find the work done by the wind as the boat moves 120 \(\mathrm{ft.}\)

Find an equation of the plane with \(x\) -intercept \(a, y\) -intercept \(b,\) and \(z\) -intercept \(c .\)

\(39-44=\) Find the derivative of the vector function. $$ \mathbf{r}(t)=\left\langle\tan t, \sec t, 1 / t^{2}\right\rangle $$

A wrench 30 \(\mathrm{cm}\) long lies along the positive \(y\) -axis and grips a bolt at the origin. A force is applied in the direction \(\langle 0,3,-4\rangle\) at the end of the wrench. Find the magnitude of the force needed to supply 100 \(\mathrm{N} \cdot \mathrm{m}\) of torque to the bolt.

Find equations of the normal plane and osculating plane of the curve at the given point. $$x=2 \sin 3 t, y=t, z=2 \cos 3 t ; \quad(0, \pi,-2)$$