Question 1

&#10; &#10;A) x = 13t, y = t, z = t&#10;B) x = 10 + t, y = t, z = t&#10;C) x = 10 + 13t, y = 9 + t, z = 1&#10;D) x = 10 + 13t, y = 9 + t, z = 0

Accepted Answer

To find the parametric equations for the tangent line at $t = t_0 = 2$, we first compute the derivative of $\mathbf{r}(t)$ to get the tangent vector at $t_0$. The derivative $\mathbf{r}'(t) = (8t - 3)\mathbf{i} + \mathbf{j}$. Evaluating this at $t_0 = 2$ gives $\mathbf{r}'(2) = (16 - 3)\mathbf{i} + \mathbf{j} = 13\mathbf{i} + \mathbf{j}$. The $z$-component of $\mathbf{r}(t)$ is constant, so its derivative is $0\mathbf{k}$, indicating no change in the $z$-direction along the tangent. The point on the curve at $t_0 = 2$ is $\mathbf{r}(2) = (16 - 6)\mathbf{i} + (2 + 7)\mathbf{j} + \mathbf{k} = 10\mathbf{i} + 9\mathbf{j} + \mathbf{k}$. The parametric equations for the tangent line are $x = 10 + 13t$, $y = 9 + t$, and $z = 1$, corresponding to the point and direction vector at $t_0$.

Question 2

The position vector of a particle is r(t). Find the requested vector.&#10; &#10;A) a(1) = -36i - j + 8k&#10;B) a(1) = -9i + 8k&#10;C) a(1) = 36i + 8k&#10;D) a(1) = -36i + 8k

Accepted Answer

The acceleration vector is given by the second derivative of the position vector:

a(t) = r''(t) = (6 - 36t^2)i - j + (8t - 4)k

So, the acceleration at t=1 is:

a(1) = (6 - 36*1^2)i - j + (8*1 - 4)k = -36i - j + 8k

Therefore, the correct answer is D.

Question 3

The position vector of a particle is r(t). Find the requested vector.&#10;

Accepted Answer

C

Question 4

The position vector of a particle is r(t). Find the requested vector.&#10;

Accepted Answer

The velocity vector $\mathbf{v}(t)$ is found by differentiating the position vector $\mathbf{r}(t)$ with respect to $t$. For $\mathbf{r}(t) = \cos(5t)\mathbf{i} + 7\ln(t-2)\mathbf{j} - \frac{t^3}{10}\mathbf{k}$, the differentiation gives $\mathbf{v}(t) = -5\sin(5t)\mathbf{i} + \frac{7}{t-2}\mathbf{j} - \frac{3t^2}{10}\mathbf{k}$. At $t=0$, the $\mathbf{i}$ and $\mathbf{k}$ components are undefined or not applicable due to the logarithmic term and the division by zero in the $\mathbf{j}$ component, indicating that the velocity cannot be directly evaluated at $t=0$ for those components. Thus, the provided options are incorrect based on the mathematical operations involved, and the correct approach to find $\mathbf{v}(0)$ would consider the limits or continuity of the function, which is not directly addressed in the options given.

Question 5

The position vector of a particle is r(t). Find the requested vector.&#10;

Accepted Answer

To find the velocity vector, we differentiate the position vector.

$$\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t) = \frac{d}{dt}(4\sec^2(t)\mathbf{i}-5	an(t)\mathbf{j}+10t^2\mathbf{k})$$$$=8\sec(t)	an(t)\mathbf{i}-5\sec^2(t)\mathbf{j}+20t\mathbf{k}$$

Then, we substitute $t=\frac{\pi}{4}$ to get the velocity at that time.

$$\mathbf{v}\left(\frac{\pi}{4}ight)=8\sec\left(\frac{\pi}{4}ight)	an\left(\frac{\pi}{4}ight)\mathbf{i}-5\sec^2\left(\frac{\pi}{4}ight)\mathbf{j}+20\left(\frac{\pi}{4}ight)\mathbf{k}$$$$=8(1)(1)\mathbf{i}-5(2)\mathbf{j}+5\pi\mathbf{k}=16\mathbf{i}-10\mathbf{j}+5\pi\mathbf{k}$$

Therefore, the correct choice is (C).

Question 6

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.&#10;

Accepted Answer

The velocity vector $\mathbf{v}(t)$ is found by differentiating the position vector $\mathbf{r}(t)$ with respect to time $t$. Differentiating each component of $\mathbf{r}(t) = (-8t^2 - 3)\mathbf{i} + \left(\frac{1}{9}t^3\right)\mathbf{j}$ gives $\mathbf{v}(t) = (-16t)\mathbf{i} + \left(\frac{1}{3}t^2\right)\mathbf{j}$, which matches choice B.

Question 7

The position vector of a particle is r(t). Find the requested vector.&#10;

Accepted Answer

The answer of The position vector of a particle is...

Question 8

The position vector of a particle is r(t). Find the requested vector.&#10;

Accepted Answer

The answer of The position vector of a particle is...

Question 9

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors&#10;at time t = 0.&#10;

Accepted Answer

The velocity vector $v(t)$ is the derivative of the position vector $r(t)$ with respect to time $t$, and the acceleration vector $a(t)$ is the derivative of the velocity vector with respect to time. For the given position vector $r(t) = (3t^2 + 3)i + (4t^3 - 2t)k$, we find the velocity and acceleration vectors as follows:1. Velocity vector $v(t) = \frac{dr(t)}{dt} = (6t)i + (12t^2 - 2)k$.2. Acceleration vector $a(t) = \frac{dv(t)}{dt} = 6i + (24t)k$.At $t = 0$, the velocity vector $v(0) = 6i - 2k$ and the acceleration vector $a(0) = 6i$.The angle $\theta$ between the velocity and acceleration vectors can be found using the dot product formula: $$v \cdot a = |v||a|\cos(\theta)$$At $t = 0$, $v(0) \cdot a(0) = (6i - 2k) \cdot (6i) = 36 - 0 = 36$, and the magnitudes are $|v(0)| = \sqrt{6^2 + (-2)^2} = \sqrt{40}$ and $|a(0)| = \sqrt{6^2} = \sqrt{36}$.Since the dot product is positive and both vectors have non-zero magnitudes, but the acceleration vector has no component in the $k$ direction while the velocity does, it indicates they are perpendicular at $t = 0$. Thus, the angle between them is $\frac{\pi}{2}$ or 90 degrees.

Question 10

The position vector of a particle is r(t). Find the requested vector.&#10; &#10;A) v(3) = 46i + 81j + 6k&#10;B) v(3) = 25i - 27j - 3k&#10;C) v(3) = 46i - 81j - 6k&#10;D) v(3) = 38i - 81j - 6k

Accepted Answer

The velocity vector can be found by taking the derivative of the position vector with respect to time. Thus, we have:

$\mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = (14t+4)\mathbf{i} -9t^2\mathbf{j} -2t\mathbf{k}$

To find the velocity at $t=3$, we simply substitute $t=3$ into this expression:

$\mathbf{v}(3) = (14(3)+4)\mathbf{i} -9(3)^2\mathbf{j} -2(3)\mathbf{k} = 46\mathbf{i}-81\mathbf{j}-6\mathbf{k}$

Therefore, the answer is C.

Question 11

The position vector of a particle is r(t). Find the requested vector.&#10;

Accepted Answer

The answer of The position vector of a particle is...

Question 12

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.&#10;

Accepted Answer

The answer of If r(t) is the position vector of...

Question 13

The position vector of a particle is r(t). Find the requested vector.&#10;

Accepted Answer

The answer of The position vector of a particle is...

Question 14

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors&#10;at time t = 0.&#10;

Accepted Answer

The answer of The vector r(t) is the position vector...

Question 15

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
Find the acceleration vector. r(t) = (cos 4t)i + (5 sin t)j

A) a = (-4 cos 4t)i + (5 sin t)j
B) a = (-16 cos 4t)i + (-5 sin t)j
C) a = (-16 cos 4t)i + (-25 sin t)j
D) a = (16 cos 4t)i + (-5 sin t)j

Accepted Answer

The answer of If r(t) is the position vector of...

Question 16

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.&#10;Find the velocity vector. r(t) = (cot t)i + (csc t)j

Accepted Answer

The answer of If r(t) is the position vector of...

Question 17

The position vector of a particle is r(t). Find the requested vector.&#10;

Accepted Answer

The answer of The position vector of a particle is...

Question 18

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors&#10;at time t = 0.&#10;

Accepted Answer

The answer of The vector r(t) is the position vector...

Question 19

If r(t) is the position vector of a particle in the plane at time t, find the indicated vector.
Find the acceleration vector. r(t) = (cos 4t)i + (5 sin t)j

A) a = (-4 cos 4t)i + (5 sin t)j
B) a = (-16 cos 4t)i + (-5 sin t)j
C) a = (-16 cos 4t)i + (-25 sin t)j
D) a = (16 cos 4t)i + (-5 sin t)j

Accepted Answer

The answer of If r(t) is the position vector of...

Question 20

The vector r(t) is the position vector of a particle at time t. Find the angle between the velocity and the acceleration vectors&#10;at time t = 0.&#10;

Accepted Answer

The answer of The vector r(t) is the position vector...

Question 21

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 22

Provide an appropriate response.&#10; &#10;A) Path (4) and Path (2)&#10;B) Path (1)&#10;C) Path (2) and Path (3)&#10;D) Path (3)

Accepted Answer

The answer of Provide an appropriate response.&#10; &#10;A) Path (4)...

Question 23

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 24

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#10; ...

Question 25

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 26

Provide an appropriate response.&#10;Show that if a particle's velocity vector is always orthogonal to its acceleration vector then the particle's speed is&#10;constant.

Accepted Answer

The answer of Provide an appropriate response.&#10;Show that if a...

Question 27

&#10; &#10;A) x = 10, y = -3t, z = -8 + t&#10;B) x = 10t, y = -3, z = 1 - t&#10;C) x = 10t, y = 3, z = 1 + t&#10;D) x = 10t, y = -3, z = 1 - 8t

Accepted Answer

The answer of  &#10; &#10;A) x = 10, y...

Question 28

Provide an appropriate response.&#10; &#10;A) Path (2) and Path (4)&#10;B) Path (4)&#10;C) Path (3)&#10;D) Path (1)

Accepted Answer

The answer of Provide an appropriate response.&#10; &#10;A) Path (2)...

Question 29

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 30

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 31

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 32

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 33

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 34

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#10; ...

Question 35

Evaluate the integral.&#10;

Accepted Answer

The answer of Evaluate the integral.&#10; ...

Question 36

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 37

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#10; ...

Question 38

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#10; ...

Question 39

Provide an appropriate response.&#10;The position of a particle is given by r(t) = sin 8t i + cos 3t j. Find the velocity vector for the particle.&#10;A) cos 8t i - sin 3t j&#10;B) 8 i + 3 j&#10;C) 8 cos 8t i - 3 sin 3t j&#10;D) 8 sin 8t i + 3 cos 3t j

Accepted Answer

The answer of Provide an appropriate response.&#10;The position of a...

Question 40

11eb49f8_e077_b2d7_b71a_0f3bc433f000_TB6591_0011eb49f8_e077_d9e9_b71a_8309705f7ad7_TB6591_00

Accepted Answer

The answer of 11eb49f8_e077_b2d7_b71a_0f3bc433f000_TB6591_0011eb49f8_e077_d9e9_b71a_8309705f7ad7_TB6591_00...

Question 41

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 33 sec
B) 35 sec
C) 37 sec
D) It will never get that far downrange.

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 42

Solve the problem.&#10;At time t = 0 a particle is located at the point (4, -3, 2). It travels in a straight line to the point (6, -2, 1), has speed 2 at (4, -3, 2) and constant acceleration 6i - j - k. Find an equation for the position vector r(t) of the particle at time t.

Accepted Answer

The answer of Solve the problem.&#10;At time t = 0...

Question 43

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#10; ...

Question 44

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 33 sec
B) 35 sec
C) 37 sec
D) It will never get that far downrange.

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 45

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 33 sec
B) 35 sec
C) 37 sec
D) It will never get that far downrange.

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 46

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#10; ...

Question 47

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 33 sec
B) 35 sec
C) 37 sec
D) It will never get that far downrange.

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 48

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 33 sec
B) 35 sec
C) 37 sec
D) It will never get that far downrange.

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 49

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#10; ...

Question 50

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 51

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface
Find the muzzle speed of a gun whose maximum range is 18.8 km. Round your answer to the nearest tenth.

A) 13.6 km/sec
B) 14.1 km/sec
C) 184.2 km/sec
D) 184.7 km/sec

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 52

Provide an appropriate response.&#10;Increasing the initial speed of a projectile by a factor of 6 increases its range by what factor? Assume the elevation is the same.&#10;A) Factor of 6&#10;B) Factor of 2.45&#10;C) The elevation must be specified before an answer can be found.&#10;D) Factor of 36

Accepted Answer

The answer of Provide an appropriate response.&#10;Increasing the initial speed...

Question 53

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#10; ...

Question 54

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 2 ft
B) 74.4 ft
C) 227.8 ft
D) 6.8 ft

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 55

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 2 ft
B) 74.4 ft
C) 227.8 ft
D) 6.8 ft

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 56

Solve the initial value problem.&#10;

Accepted Answer

The answer of Solve the initial value problem.&#10; ...

Question 57

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface

A) 33 sec
B) 35 sec
C) 37 sec
D) It will never get that far downrange.

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 58

Provide an appropriate response.&#10;What two angles of elevation will enable a projectile to reach a target 17 km downrange on the same level as the gun if the projectile's initial speed is 420 m/sec? Assume there is no wind resistance.

Accepted Answer

The answer of Provide an appropriate response.&#10;What two angles of...

Question 59

Provide an appropriate response.&#10;

Accepted Answer

The answer of Provide an appropriate response.&#10; ...

Question 60

Solve the problem. Unless stated otherwise, assume that the projectile flight is ideal, that the launch angle is measured
from the horizontal, and that the projectile is launched from the origin over a horizontal surface
Find the muzzle speed of a gun whose maximum range is 18.8 km. Round your answer to the nearest tenth.

A) 13.6 km/sec
B) 14.1 km/sec
C) 184.2 km/sec
D) 184.7 km/sec

Accepted Answer

The answer of Solve the problem. Unless stated otherwise, assume...

Question 61

Calculate the arc length of the indicated portion of the curve r(t).&#10; &#10;A) -1089&#10;B) -99&#10;C) 121&#10;D) 1331

Accepted Answer

The answer of Calculate the arc length of the indicated...

Question 62

Find the unit tangent vector of the given curve.&#10; &#10;A) T = (5 sin 9t)i - (5 cos 9t)j&#10;B) T = (sin 9t)i - (cos 9t)j&#10;C) T = (135 sin 9t)i -(135 cos 9t)j&#10;D) T = (5 cos 9t)i - (5 sin 9t)j

Accepted Answer

The answer of Find the unit tangent vector of the...

Question 63

Calculate the arc length of the indicated portion of the curve r(t).&#10; &#10;A) -35&#10;B) 45&#10;C) -175&#10;D) 225

Accepted Answer

The answer of Calculate the arc length of the indicated...

Question 64

Provide an appropriate response.&#10; &#10;A) Maximum height = 3.48 feet;&#10;B) Maximum height = 0.21 feet; time to maximum height = 0.05 sec time to maximum height = -0.21 sec&#10;C) Maximum height = 3.51 feet;&#10;D) Maximum height = 8.71 feet; time to maximum height = 0.02 sec time to maximum height = 0.74 sec

Accepted Answer

The answer of Provide an appropriate response.&#10; &#10;A) Maximum height...

Deck 14: Vector-Valued Functions and Motion in Space