Question 1

Determine whether the pair of vectors is orthogonal.
-$$3 \mathbf { i } - 3 \mathbf { j } , - 8 \mathbf { i } - 8 \mathbf { j }$$

Accepted Answer

The dot product of the two vectors is 0, so they are orthogonal.

Question 2

Solve the problem.
-One rope pulls a barge directly east with a force of 68 newtons, and another rope pulls the barge directly north with a force of 62 newtons. Find the magnitude of the resultant force acting on the barge.

Accepted Answer

The magnitude of the resultant force can be found using the Pythagorean theorem since the forces are perpendicular to each other: $\sqrt{68^2 + 62^2} \approx 92$ newtons.

Question 3

Solve the problem.
-A wind has an easterly component of $6 \mathrm {~km} / \mathrm { h }$ and a northerly component of $17 \mathrm {~km} / \mathrm { h }$. Find the direction of the wind.

Accepted Answer

The answer of Solve the problem.
-A wind has an easterly...

Question 4

Solve the problem.
-What is the minimum force required to prevent a ball weighing $26.6 \mathrm { lb }$ from rolling down a ramp inclined $12.4 ^ { \circ }$ with the horizontal?

Accepted Answer

The minimum force required to prevent the ball from rolling down the ramp can be found by calculating the component of the ball's weight parallel to the ramp's surface. This force can be calculated using the formula $F = mg\sin(\theta)$, where $m$ is the mass of the ball, $g$ is the acceleration due to gravity (approximately $32.2 \mathrm{ft/s^2}$ for objects near the Earth's surface), and $\theta$ is the angle of the incline.First, convert the weight of the ball from pounds to mass in slugs using the relation $1 \mathrm{lb} = 1 \mathrm{slug} \cdot 32.2 \mathrm{ft/s^2}$:$$26.6 \mathrm{lb} = m \cdot 32.2 \mathrm{ft/s^2}$$$$m = \frac{26.6}{32.2} \mathrm{slugs}$$Then, calculate the force:$$F = mg\sin(\theta) = \frac{26.6}{32.2} \cdot 32.2 \cdot \sin(12.4^\circ)$$$$F \approx 5.7 \mathrm{lb}$$Therefore, the minimum force required to prevent the ball from rolling down the ramp is approximately $5.7 \mathrm{lb}$.

Question 5

Determine whether the pair of vectors is orthogonal.
-$$3 \mathbf { i } - 8 \mathbf { j } , - 8 \mathbf { i } + 3 \mathbf { j }$$

Accepted Answer

The answer of Determine whether the pair of vectors is...

Question 6

Solve the problem.
-A pilot wants to fly on a bearing of $60.5 ^ { \circ }$. By flying due east, he finds that a 57 -mph wind, blowing from the south, puts him on course. Find the airspeed of the plane.

Accepted Answer

The airspeed of the plane can be found using vector addition, where the eastward component of the plane's velocity and the wind's velocity vector combine to give the resultant velocity vector on a bearing of $60.5^\circ$. Since the wind is blowing from the south (directly towards the north) and affects the plane's eastward course, we can form a right-angled triangle with the wind speed as one leg (57 mph), the plane's airspeed as the hypotenuse, and the eastward component of the plane's velocity as the other leg. The angle between the plane's desired course (bearing of $60.5^\circ$) and the east direction is $60.5^\circ$. Using trigonometry, specifically the sine function, $\sin(60.5^\circ) = \frac{57}{\text{airspeed}}$. Solving for the airspeed gives approximately $101 \mathrm { mph }$, making option B correct.

Question 7

Solve the problem.
-Two people are carrying a box. One person exerts a force of 129 pounds at an angle of $46.2 ^ { \circ }$ with the horizontal. The other person exerts a force of 93 pounds at an angle of $52.5 ^ { \circ }$. Find the weight of the box.

Accepted Answer

The answer of Solve the problem.
-Two people are carrying a...

Question 8

Solve the problem.
-A force of $35 \mathrm { lb }$ is required to hold a $78 \mathrm { lb }$ toolbox on an incline. What angle does the incline make with the horizontal?

Accepted Answer

The angle can be found using the formula for the component of the weight parallel to the incline, which is $F = W \sin(\theta)$, where $F$ is the force parallel to the incline (35 lb), $W$ is the weight of the object (78 lb), and $\theta$ is the angle of the incline. Solving for $\theta$, we get $\theta = \sin^{-1}(F/W) = \sin^{-1}(35/78) \approx 26.7^\circ$.

Question 9

Solve the problem.
-Two forces, of $67.7$ and $15.8 \mathrm { lb }$, forming an angle of $26.7 ^ { \circ }$, act at a point in the plane. Find the magnitude of the resultant force.

Accepted Answer

The magnitude of the resultant force can be found using the formula for the resultant of two forces at an angle: $R = \sqrt{F_1^2 + F_2^2 + 2F_1F_2\cos(\theta)}$, where $F_1$ and $F_2$ are the magnitudes of the two forces and $\theta$ is the angle between them. Substituting the given values, $R = \sqrt{67.7^2 + 15.8^2 + 2(67.7)(15.8)\cos(26.7^\circ)}$, which calculates to approximately $82.1 \mathrm { lb }$.

Question 10

Solve the problem.
-A pilot wants to fly on a bearing of $60.1 ^ { \circ }$. By flying due east, he finds that a $51 - \mathrm { mph }$ wind, blowing from the south, puts him on course. Find the ground speed of the plane.

Accepted Answer

The answer of Solve the problem.
-A pilot wants to fly...

Question 11

Solve the problem.
-Suppose you would like to cross a 216 -foot wide river in a boat. Assume that the boat can travel $38 \mathrm { mph }$ relative to the water and that the current is flowing west at the rate of $6 \mathrm { mph }$. If the bearing is chosen so that the boat will land at a point exactly across from its starting point, how long will it take for the boat to make the crossing? Give your answer to the nearest second.

Accepted Answer

The answer of Solve the problem.
-Suppose you would like to...

Question 12

Solve the problem.
-A force of $670 \mathrm { lb }$ is required to pull a boat up a ramp inclined at $25 ^ { \circ }$ with the horizontal. How much does the boat weigh?

Accepted Answer

The weight of the boat can be found by dividing the force required to pull the boat up the ramp by the cosine of the ramp's angle with the horizontal. This is because the force required to pull the boat is the component of the boat's weight parallel to the ramp. The weight of the boat is the hypotenuse of the right triangle formed by the weight components parallel and perpendicular to the ramp. Therefore, weight = $670 \mathrm { lb } / \cos(25^\circ) \approx 1585 \mathrm { lb }$.

Question 13

Solve the problem.
-Two boats are pulling a disabled vessel toward the landing dock with forces of $930 \mathrm { lb }$ and $890 \mathrm { lb }$. The angle between the forces is $28.6 ^ { \circ }$. Find the direction and magnitude of the equilibrant.

Accepted Answer

The equilibrant force is equal in magnitude but opposite in direction to the resultant force of the two pulling forces. To find the magnitude of the resultant force, we use the law of cosines because we have two sides of a triangle (the forces) and the included angle. The magnitude $R$ of the resultant force is given by $R = \sqrt{930^2 + 890^2 - 2(930)(890)\cos(28.6^\circ)}$. Calculating this gives a magnitude close to $1764 \, \text{lb}$. The direction of the equilibrant is directly opposite to the resultant, making it $180^\circ - 28.6^\circ = 151.4^\circ$ from the smaller force (890 lb), but since the answer choices are in relation to the 930 lb force, and the angle between the forces is $28.6^\circ$, the direction relative to the 930 lb force is not directly provided in the options. However, understanding that the equilibrant is opposite to the resultant, and given the context of the options, the closest match is option A, which suggests a significant angle with the 930 lb force, aligning with the concept of the equilibrant being opposite in direction to the resultant. The discrepancy in the explanation of the angle arises from interpreting the provided options; the magnitude is the primary indicator for the correct answer.

Question 14

Solve the problem.
-A ship leaves point $A$ and travels directly north to point $B$. From point $B$, the ship then travels due east to point $C$. The magnitude of the vector from point $A$ to point $C$ is 134 . Find the magnitude of the northern vector. Find the magnitude of the eastern vector.

Accepted Answer

The vector from A to C forms a right-angled triangle with the north and east vectors as the legs. Since the magnitude of the vector from A to C is 134, and it forms a 45-degree angle (implied by the equal magnitudes of the north and east vectors), the north and east vectors must each have a magnitude of $67 \sqrt { 2 }$.

Question 15

Solve the problem.
-A box weighing $77 \mathrm { lb }$ is hanging from the end of a rope. The box is pulled sideways by a horizontal rope with a force of $20 \mathrm { lb }$. What angle, to the nearest degree, does the first rope make with the vertical?

Accepted Answer

The answer of Solve the problem.
-A box weighing \(77 \mathrm...

Question 16

Solve the problem.
-Two forces of 575 newtons and 271 newtons act at a point. The resultant force is 657 newtons. Find the angle between the forces.

Accepted Answer

The answer of Solve the problem.
-Two forces of 575 newtons...

Question 17

Solve the problem.
-Starting at point A, a ship sails $50 \mathrm {~km}$ on a bearing of $243 ^ { \circ }$, then turns and sails $71 \mathrm {~km}$ on a bearing of $299 ^ { \circ }$. Find the distance of the ship from point A.

Accepted Answer

Using the law of cosines, the distance from point A is calculated as $\sqrt{50^2 + 71^2 - 2 \times 50 \times 71 \times \cos(56^\circ)} \approx 107 \mathrm{~km}$.

Question 18

Solve the problem.
-A plane is heading due south with an airspeed of $276 \mathrm { mph }$. A wind from a direction of $52 ^ { \circ }$ is blowing at $17 \mathrm { mph }$. Find the bearing of the plane.

Accepted Answer

The plane's velocity vector is due south, and the wind vector is at $52^\circ$ from the north. Using vector addition, the resultant vector's direction can be calculated, resulting in a bearing of $183^\circ$.

Question 19

Solve the problem.
-A hot-air balloon is rising vertically $13 \mathrm { ft } / \mathrm { sec }$ while the wind is blowing horizontally at $6 \mathrm { ft } / \mathrm { sec }$. Find the angle that the balloon makes with the horizontal.

Accepted Answer

The angle $\theta$ that the balloon makes with the horizontal can be found using the tangent function, where $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{13}{6}$. Solving for $\theta$ gives $\theta \approx 65.2^\circ$.

Question 20

Solve the problem.
-Suppose you would like to cross a 202-foot wide river in a boat. Assume that the boat can travel $39 \mathrm { mph }$ relative to the water and that the current is flowing west at the rate of $6 \mathrm { mph }$. What bearing should be chosen so that the boat will land at a point exactly across from its starting point? Give your answer to the nearest tenth of a degree.

Accepted Answer

The answer of Solve the problem.
-Suppose you would like to...

Quiz 8: Applications of Trigonometry

Determine whether the pair of vectors is orthogonal. - $3 \mathbf { i } - 3 \mathbf { j } , - 8 \mathbf { i } - 8 \mathbf { j }$

Solve the problem. -One rope pulls a barge directly east with a force of 68 newtons, and another rope pulls the barge directly north with a force of 62 newtons. Find the magnitude of the resultant force acting on the barge.

Solve the problem. -A wind has an easterly component of $6 \mathrm {~km} / \mathrm { h }$ and a northerly component of $17 \mathrm {~km} / \mathrm { h }$ . Find the direction of the wind.

Solve the problem. -What is the minimum force required to prevent a ball weighing $26.6 \mathrm { lb }$ from rolling down a ramp inclined $12.4 ^ { \circ }$ with the horizontal?

Determine whether the pair of vectors is orthogonal. - $3 \mathbf { i } - 8 \mathbf { j } , - 8 \mathbf { i } + 3 \mathbf { j }$

Solve the problem. -A pilot wants to fly on a bearing of $60.5 ^ { \circ }$ . By flying due east, he finds that a 57 -mph wind, blowing from the south, puts him on course. Find the airspeed of the plane.

Solve the problem. -Two people are carrying a box. One person exerts a force of 129 pounds at an angle of $46.2 ^ { \circ }$ with the horizontal. The other person exerts a force of 93 pounds at an angle of $52.5 ^ { \circ }$ . Find the weight of the box.

Solve the problem. -A force of $35 \mathrm { lb }$ is required to hold a $78 \mathrm { lb }$ toolbox on an incline. What angle does the incline make with the horizontal?

Solve the problem. -Two forces, of $67.7$ and $15.8 \mathrm { lb }$ , forming an angle of $26.7 ^ { \circ }$ , act at a point in the plane. Find the magnitude of the resultant force.

Solve the problem. -A pilot wants to fly on a bearing of $60.1 ^ { \circ }$ . By flying due east, he finds that a $51 - \mathrm { mph }$ wind, blowing from the south, puts him on course. Find the ground speed of the plane.

Solve the problem. -A force of $670 \mathrm { lb }$ is required to pull a boat up a ramp inclined at $25 ^ { \circ }$ with the horizontal. How much does the boat weigh?

Solve the problem. -Two boats are pulling a disabled vessel toward the landing dock with forces of $930 \mathrm { lb }$ and $890 \mathrm { lb }$ . The angle between the forces is $28.6 ^ { \circ }$ . Find the direction and magnitude of the equilibrant.

Solve the problem. -A ship leaves point $A$ and travels directly north to point $B$ . From point $B$ , the ship then travels due east to point $C$ . The magnitude of the vector from point $A$ to point $C$ is 134 . Find the magnitude of the northern vector. Find the magnitude of the eastern vector.

Solve the problem. -A box weighing $77 \mathrm { lb }$ is hanging from the end of a rope. The box is pulled sideways by a horizontal rope with a force of $20 \mathrm { lb }$ . What angle, to the nearest degree, does the first rope make with the vertical?

Solve the problem. -Two forces of 575 newtons and 271 newtons act at a point. The resultant force is 657 newtons. Find the angle between the forces.

Solve the problem. -Starting at point A, a ship sails $50 \mathrm {~km}$ on a bearing of $243 ^ { \circ }$ , then turns and sails $71 \mathrm {~km}$ on a bearing of $299 ^ { \circ }$ . Find the distance of the ship from point A.

Solve the problem. -A plane is heading due south with an airspeed of $276 \mathrm { mph }$ . A wind from a direction of $52 ^ { \circ }$ is blowing at $17 \mathrm { mph }$ . Find the bearing of the plane.

Solve the problem. -A hot-air balloon is rising vertically $13 \mathrm { ft } / \mathrm { sec }$ while the wind is blowing horizontally at $6 \mathrm { ft } / \mathrm { sec }$ . Find the angle that the balloon makes with the horizontal.

Quiz 8: Applications of Trigonometry

Determine whether the pair of vectors is orthogonal. - 3i−3j,−8i−8j3 \mathbf { i } - 3 \mathbf { j } , - 8 \mathbf { i } - 8 \mathbf { j }3i−3j,−8i−8j

Solve the problem. -One rope pulls a barge directly east with a force of 68 newtons, and another rope pulls the barge directly north with a force of 62 newtons. Find the magnitude of the resultant force acting on the barge.

Solve the problem. -A wind has an easterly component of 6 km/h6 \mathrm {~km} / \mathrm { h }6 km/h and a northerly component of 17 km/h17 \mathrm {~km} / \mathrm { h }17 km/h . Find the direction of the wind.

Solve the problem. -What is the minimum force required to prevent a ball weighing 26.6lb26.6 \mathrm { lb }26.6lb from rolling down a ramp inclined 12.4∘12.4 ^ { \circ }12.4∘ with the horizontal?

Determine whether the pair of vectors is orthogonal. - 3i−8j,−8i+3j3 \mathbf { i } - 8 \mathbf { j } , - 8 \mathbf { i } + 3 \mathbf { j }3i−8j,−8i+3j

Solve the problem. -A pilot wants to fly on a bearing of 60.5∘60.5 ^ { \circ }60.5∘ . By flying due east, he finds that a 57 -mph wind, blowing from the south, puts him on course. Find the airspeed of the plane.

Solve the problem. -Two people are carrying a box. One person exerts a force of 129 pounds at an angle of 46.2∘46.2 ^ { \circ }46.2∘ with the horizontal. The other person exerts a force of 93 pounds at an angle of 52.5∘52.5 ^ { \circ }52.5∘ . Find the weight of the box.

Solve the problem. -A force of 35lb35 \mathrm { lb }35lb is required to hold a 78lb78 \mathrm { lb }78lb toolbox on an incline. What angle does the incline make with the horizontal?

Solve the problem. -Two forces, of 67.767.767.7 and 15.8lb15.8 \mathrm { lb }15.8lb , forming an angle of 26.7∘26.7 ^ { \circ }26.7∘ , act at a point in the plane. Find the magnitude of the resultant force.

Solve the problem. -A pilot wants to fly on a bearing of 60.1∘60.1 ^ { \circ }60.1∘ . By flying due east, he finds that a 51−mph51 - \mathrm { mph }51−mph wind, blowing from the south, puts him on course. Find the ground speed of the plane.

Solve the problem. -A force of 670lb670 \mathrm { lb }670lb is required to pull a boat up a ramp inclined at 25∘25 ^ { \circ }25∘ with the horizontal. How much does the boat weigh?

Solve the problem. -Two boats are pulling a disabled vessel toward the landing dock with forces of 930lb930 \mathrm { lb }930lb and 890lb890 \mathrm { lb }890lb . The angle between the forces is 28.6∘28.6 ^ { \circ }28.6∘ . Find the direction and magnitude of the equilibrant.

Solve the problem. -A ship leaves point AAA and travels directly north to point BBB . From point BBB , the ship then travels due east to point CCC . The magnitude of the vector from point AAA to point CCC is 134 . Find the magnitude of the northern vector. Find the magnitude of the eastern vector.

Solve the problem. -A box weighing 77lb77 \mathrm { lb }77lb is hanging from the end of a rope. The box is pulled sideways by a horizontal rope with a force of 20lb20 \mathrm { lb }20lb . What angle, to the nearest degree, does the first rope make with the vertical?