Chapter 7

Work Mark pulls Allison and Mattie in a wagon by exerting a force of 25 pounds on the handle at an angle of \(30^{\circ}\) with the horizontal (Figure 7 ). How much work is done by Mark in pulling the wagon 350 fect?

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(3 \mathbf{U}+2 \mathbf{V}\). $$\mathbf{U}=5 \mathbf{i}+3 \mathbf{j}, \mathbf{V}=-3 \mathbf{i}-5 \mathbf{j}$$

Find \(c\) for triangle \(A B C\) if \(A=43^{\circ}, B=12^{\circ}\), and \(b=25\) centimeters. a. \(98 \mathrm{~cm}\) b. \(34 \mathrm{~cm}\) c. \(120 \mathrm{~cm}\) d. \(82 \mathrm{~cm}\)

If \(\mathbf{U}=7 \mathbf{i}+2 \mathbf{j}\) and \(\mathbf{V}=3 \mathbf{i}-4 \mathrm{j}\), find \(\mathbf{U} \cdot \mathbf{V}\). a. \(21 \mathbf{i}-8 \mathbf{j}\) b. \(21 \mathrm{i}^{2}-22 \mathrm{ij}-8 \mathrm{j}^{2}\) c. 2 d. 13

Vector \(\mathbf{V}\) is in standard position and makes an angle of \(30^{\circ}\) with the positive \(x\)-axis. Its magnitude is 18 . Write \(\mathbf{V}\) in component form \(\langle a, b\rangle\) and in vector component form \(a \mathbf{i}+b \mathbf{j}\).

Gina is standing near a building and notices that the angle of elevation to the top of the building is \(68^{\circ}\). She then walks 72 feet further away from the building and notices that the angle of elevation to the top of the building is now only \(51^{\circ}\). Find the height of the building. a. \(25 \mathrm{ft}\) b. \(56 \mathrm{ft}\) c. \(149 \mathrm{ft}\) d. \(180 \mathrm{ft}\)

Find the angle between \(\mathbf{U}=3 \mathbf{i}+7 \mathbf{j}\) and \(\mathbf{V}=\mathbf{i}-4 \mathbf{j}\). a. \(52.8^{\circ}\) b. \(142.8^{\circ}\) c. \(137.2^{\circ}\) d. \(157.9^{\circ}\)

Vector \(\mathbf{U}\) is in standard position and makes an angle of \(120^{\circ}\) with the positive \(x\)-axis. Its magnitude is 24 . Write \(\mathbf{U}\) in component form \(\langle a, b\rangle\) and in vector component form \(a \mathbf{i}+b \mathbf{j}\).

The problems that follow review material we covered in Section 6.3. Find all solutions in radians using exact values only. $$ \sin 3 x=1 / 2 $$

Which pair of vectors are perpendicular? a. \(3 \mathbf{i}+4 \mathbf{j}\) and \(8 \mathbf{i}-6 \mathbf{j}\) b. \(3 \mathbf{i}+2 \mathbf{j}\) and \(2 \mathbf{i}+3 \mathbf{j}\) c. \(\mathbf{i}+5 \mathbf{j}\) and \(\mathbf{i}-5 \mathbf{j}\) d. \(2 \mathbf{i}-5 \mathbf{j}\) and \(-7 \mathbf{i}-3 \mathbf{j}\)

Vector \(\mathbf{W}\) is in standard position and makes an angle of \(270^{\circ}\) with the positive \(x\)-axis. Its magnitude is 8 . Write \(\mathbf{W}\) in component form \(\langle a, b\rangle\) and in vector component form \(a \mathbf{i}+b \mathbf{j}\).

The problems that follow review material we covered in Section 6.3. Find all solutions in radians using exact values only. $$ \cos 4 x=-\frac{1}{2} $$

Find the work performed when a force \(\mathbf{F}=15 \mathbf{i}-9 \mathbf{j}\) is applied to an object whose resulting motion is represented by displacement vector \(\mathbf{d}=80 \mathrm{i}+12 \mathrm{j}\). Assume the force is measured in pounds and the displacement in feet. a. \(1,415 \mathrm{ft}-\mathrm{lb}\) b. \(552 \mathrm{ft}-\mathrm{lb}\) c. \(1,092 \mathrm{ft}-\mathrm{lb}\) d. \(1,308 \mathrm{ft}-\mathrm{lb}\)

Vector \(\mathbf{F}\) is in standard position and makes an angle of \(315^{\circ}\) with the positive \(x\)-axis. Its magnitude is 30 . Write \(\mathbf{F}\) in component form \(\langle a, b\rangle\) and in vector component form \(a \mathbf{i}+b \mathbf{j}\).

The problems that follow review material we covered in Section 6.3. Find all solutions in radians using exact values only. $$ \tan ^{2} 3 x=1 $$

Assume vector \(\mathbf{V}\) is in standard position, has the given magnitude, and that \(\theta\) is the angle \(\mathbf{V}\) makes with the positive \(x\)-axis. Write \(\mathbf{V}\) in vector component form \(a \mathbf{i}+b \mathbf{j}\), and approximate your values to two significant digits. $$|\mathbf{V}|=5.8, \theta=71^{\circ}$$

The problems that follow review material we covered in Section 6.3. Find all solutions in radians using exact values only. $$ \tan ^{2} 4 x=3 $$

Assume vector \(\mathbf{V}\) is in standard position, has the given magnitude, and that \(\theta\) is the angle \(\mathbf{V}\) makes with the positive \(x\)-axis. Write \(\mathbf{V}\) in vector component form \(a \mathbf{i}+b \mathbf{j}\), and approximate your values to two significant digits. $$|\mathbf{V}|=8.5, \theta=97^{\circ}$$

Find all degree solutions. $$ 2 \cos ^{2} 3 \theta-9 \cos 3 \theta+4=0 $$

Assume vector \(\mathbf{V}\) is in standard position, has the given magnitude, and that \(\theta\) is the angle \(\mathbf{V}\) makes with the positive \(x\)-axis. Write \(\mathbf{V}\) in vector component form \(a \mathbf{i}+b \mathbf{j}\), and approximate your values to two significant digits. $$|\mathbf{V}|=0.55, \theta=195^{\circ}$$

Find all degree solutions. $$ 3 \sin ^{2} 2 \theta-2 \sin 2 \theta-5=0 $$

Assume vector \(\mathbf{V}\) is in standard position, has the given magnitude, and that \(\theta\) is the angle \(\mathbf{V}\) makes with the positive \(x\)-axis. Write \(\mathbf{V}\) in vector component form \(a \mathbf{i}+b \mathbf{j}\), and approximate your values to two significant digits. $$|\mathbf{V}|=330, \theta=340^{\circ}$$

Find all degree solutions. $$ \sin 4 \theta \cos 2 \theta+\cos 4 \theta \sin 2 \theta=-1 $$

Find the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. $$\mathbf{U}=\langle 3,3\rangle$$

Find all degree solutions. $$ \cos 3 \theta \cos 2 \theta-\sin 3 \theta \sin 2 \theta=-1 $$

Find the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. $$\mathbf{V}=\langle 5,-5\rangle$$

Solve each equation for \(\theta\) if \(0^{\circ} \leq \theta<360^{\circ}\). \(\sin \theta+\cos \theta=1\)

Find the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. $$\mathbf{W}=-\mathbf{i}-\sqrt{3} \mathbf{j}$$

Solve each equation for \(\theta\) if \(0^{\circ} \leq \theta<360^{\circ}\). $$ \sin \theta-\cos \theta=0 $$

Find the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. $$\mathbf{F}=-2 \sqrt{3} \mathbf{i}+2 \mathbf{j}$$

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Find \(c\) for triangle \(A B C\) if \(a=6.8\) meters, \(b=8.4\) meters, and \(C=48^{\circ}\). a. \(5.6 \mathrm{~m}\) b. \(40 \mathrm{~m}\) c. \(8.6 \mathrm{~m}\) d. \(6.4 \mathrm{~m}\)

Approximate the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. Round your answers to the nearest tenth. $$\mathrm{U}=3 \mathrm{i}+5 \mathbf{j}$$

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Find \(B\) for triangle \(A B C\) if \(a=13.8\) yards, \(b=22.3\) yards, and \(c=9.50\) yards. a. \(34.3^{\circ}\) b. \(145.7^{\circ}\) C. \(13.8^{\circ}\) d. \(160^{\circ}\)

Approximate the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. Round your answers to the nearest tenth. $$V=-i+4 j$$

Approximate the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. Round your answers to the nearest tenth. $$\mathbf{W}=\sqrt{5} \mathbf{i}-2 \mathbf{j}$$

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. A plane with an airspeed of 560 miles per hour and traveling at a heading of \(130^{\circ}\) encounters a 65 mile per hour wind blowing in the direction \(\mathrm{N} \mathrm{} 45^{\circ} \mathrm{E}\). Find the resulting ground speed of the plane. a. \(564 \mathrm{mph}\) b. \(569 \mathrm{mph}\) c. \(553 \mathrm{mph}\) d. \(557 \mathrm{mph}\)

Approximate the magnitude of each vector and the angle \(\theta, 0^{\circ} \leq \theta<360^{\circ}\), that the vector makes with the positive \(x\)-axis. Round your answers to the nearest tenth. $$F=-6 i-\sqrt{3} \mathbf{j}$$

Force After they are finished swinging, Tyler and Kelly decide to rollerskate. They come to a hill that is inclined at \(8.5^{\circ}\). Tyler pushes Kelly halfway up the hill and then holds her there (Figure 14). If Kelly weighs \(58.0\) pounds, find the magnitude of the force Tyler must push with to keep Kelly from rolling down the hill. (We are assuming that the rollerskates make the hill into a frictionless surface so that the only force keeping Kelly from rolling backwards down the hill is the force Tyler is pushing with.)

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Draw \(\mathbf{V}=\langle 2,3\rangle\) in standard position.

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Find the magnitude of vector \(\mathbf{V}=5 \mathbf{i}-2 \mathbf{j}\). a. 10 b. \(\sqrt{29}\) c. 7 d. \(\sqrt{21}\)

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. If \(\mathbf{U}=3 \mathbf{i}+7 \mathbf{j}\) and \(\mathbf{V}=\mathbf{i}-4 \mathbf{j}\), find \(\mathbf{U}+2 \mathbf{V}\). a. \(\mathbf{i}+15 \mathbf{j}\) b. \(5 \mathbf{i}+3 \mathbf{j}\) c. \(10 \mathrm{i}-6 \mathrm{j}\) d. \(5 \mathbf{i}-\mathbf{j}\)