Problem 38
Solve the given initial value problems. Find \(\vec{r}(t),\) given that \(\vec{r}^{\prime}(t)=\langle 1 /(t+1), \tan t\rangle\) and \(\vec{r}(0)=\langle 1,2\rangle\)
Problem 39
Ask you to solve a variety of problems based on the principles of projectile motion. A hunter aims at a deer which is 40 yards away. Her crossbow is at a height of \(5 \mathrm{ft}\), and she aims for a spot on the deer \(4 \mathrm{ft}\) above the ground. The crossbow fires her arrows at \(300 \mathrm{ft} / \mathrm{s}\) (a) At what angle of elevation should she hold the crossbow to hit her target? (b) If the deer is moving perpendicularly to her line of sight at a rate of \(20 \mathrm{mph}\), by approximately how much should she lead the deer in order to hit it in the desired location?
Problem 39
Find \(\vec{r}(t),\) given that \(\vec{r}^{\prime}(t)=\langle 1 /(t+1), \tan t\rangle\) and \(\vec{r}(0)=\langle 1,2\rangle\)Find \(\vec{r}(t),\) given that \(\vec{r}^{\prime \prime}(t)=\left\langle t^{2}, t, 1\right\rangle\) \(\vec{r}^{\prime}(0)=\langle 1,2,3\rangle\) and \(\vec{r}(0)=\langle 4,5,6\rangle\)
Problem 40
Solve the given initial value problems. Find \(\vec{r}(t),\) given that \(\vec{r}^{\prime \prime}(t)=\left\langle\cos t, \sin t, e^{t}\right\rangle,\) \(\vec{r}^{\prime}(0)=\langle 0,0,0\rangle\) and \(\vec{r}(0)=\langle 0,0,0\rangle\)
Problem 40
Ask you to solve a variety of problems based on the principles of projectile motion. A baseball player hits a ball at \(100 \mathrm{mph}\), with an initial height of \(3 \mathrm{ft}\) and an angle of elevation of \(20^{\circ}\), at Boston's Fenway Park. The ball flies towards the famed "Green Monster," a wall \(37 f t\) high located \(310 f t\) from home plate. (a) Show that as hit, the ball hits the wall. (b) Show that if the angle of elevation is \(21^{\circ}\), the ball clears the Green Monster.
Problem 41
Ask you to solve a variety of problems based on the principles of projectile motion. A Cessna flies at \(1000 f t\) at \(150 m p h\) and drops a box of supplies to the professor (and his wife) on an island. Ignoring wind resistance, how far horizontally will the supplies travel before they land?
Problem 41
Find the arc length of \(\vec{r}(t)\) on the indicated interval. $$ \vec{r}(t)=\langle 2 \cos t, 2 \sin t, 3 t\rangle \text { on }[0,2 \pi] $$
Problem 42
Find the arc length of \(\vec{r}(t)\) on the indicated interval. $$ \vec{r}(t)=\langle 5 \cos t, 3 \sin t, 4 \sin t\rangle \text { on }[0,2 \pi] $$
Problem 42
Ask you to solve a variety of problems based on the principles of projectile motion. A football quarterback throws a pass from a height of \(6 \mathrm{ft}\), intending to hit his receiver 20 yds away at a height of \(5 \mathrm{ft}\). (a) If the ball is thrown at a rate of \(50 \mathrm{mph}\), what angle of elevation is needed to hit his intended target? (b) If the ball is thrown at with an angle of elevation of \(8^{\circ},\) what initial ball speed is needed to hit his target?
Problem 43
Find the arc length of \(\vec{r}(t)\) on the indicated interval. $$ \vec{r}(t)=\left\langle t^{3}, t^{2}, t^{3}\right\rangle \text { on }[0,1] $$
