Chapter 7

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(2 \mathbf{U}-3 \mathbf{V}\). $$\mathbf{U}=\langle 4,4\rangle, \mathbf{V}=\langle 4,-4\rangle$$

Force A tightrope walker weighing 145 pounds is standing still at the center of a tightrope that is \(46.5\) feet long. The weight of the walker causes the center of the tightrope to move down \(14.5\) inches. Find the magnitude of the tension in the tightrope toward each end of the tightrope.

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. True Course and Speed A plane is flying with an airspeed of 244 miles per hour with heading \(272.7^{\circ}\). The wind currents are running at a constant \(45.7\) miles per hour in the direction \(262.6^{\circ}\). Find the ground speed and true course of the plane.

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=-6 \mathbf{i}+19 \mathbf{j}, \mathbf{d}=8 \mathbf{i}+55 \mathbf{j}\)

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(2 \mathbf{U}-3 \mathbf{V}\). $$\mathbf{U}=\langle-3,5\rangle, \mathbf{V}=\langle 3,-1\rangle$$

The problems that follow review material we covered in Section 6.2. Find all solutions in the interval \(0^{\circ} \leq \theta<360^{\circ}\). If rounding is necessary, round to the nearest tenth of a degree. $$7 \sin ^{2} \theta-9 \cos 2 \theta=0$$

Force If you have ever ridden on a chair lift at a ski area and had it stop, you know that the chair will pull down on the cable, dropping you down to a lower height than when the chair is in motion. Figure 19 shows a gondola that is stopped. Find the magnitude of the tension in the cable toward each end of the cable if the total weight of the gondola and its occupants is 1,850 pounds.

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. Speed and Direction A plane has an airspeed of 195 miles per hour and a heading of \(30.0^{\circ}\). The ground speed of the plane is 207 miles per hour, and its true course in in the direction of \(34.0^{\circ}\). Find the speed and direction of the air currents, assuming they are constants.

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=-67 \mathbf{i}+59 \mathbf{j}, \mathbf{d}=-96 \mathbf{i}-28 \mathbf{j}\)

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(2 \mathbf{U}-3 \mathbf{V}\). $$\mathbf{U}=\langle-5,0\rangle, \mathbf{V}=\langle 0,1\rangle$$

Find all radian solutions using exact values only. $$2 \cos x-\sec x+\tan x=0$$

Force A chair lift at a ski resort is stopped halfway between two poles that support the cable to which the chair is attached. The poles are 215 feet apart and the combined weight of the chair and the three people on the chair is 725 pounds. If the weight of the chair and the people riding it causes the chair to move to a position \(15.8\) feet below the horizontal line that connects the top of the two poles, find the tension in the cable toward each end of the cable.

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. Speed and Direction The airspeed and heading of a plane are 140 miles per hour and \(130^{\circ}\), respectively. If the ground speed of the plane is 135 miles per hour and its true course is \(137^{\circ}\), find the speed and direction of the wind currents, assuming they are constants.

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=-67 \mathbf{i}+59 \mathbf{j}, \mathbf{d}=-96 \mathbf{i}-28 \mathbf{j}\)

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(2 \mathbf{U}-3 \mathbf{V}\). $$\mathbf{U}=\langle 2,0\rangle, \mathbf{V}=\langle 0,-7\rangle$$

Find the area of triangle \(A B C\) if \(a=73.6\) millimeters, \(b=41.5\) millimeters, and \(C=22.3^{\circ}\). a. \(1,160 \mathrm{~mm}^{2}\) b. \(1,412 \mathrm{~mm}^{2}\) c. \(902 \mathrm{~mm}^{2}\) d. \(580 \mathrm{~mm}^{2}\)

Find all radian solutions using exact values only. $$2 \cos ^{2} x-\sin x=1$$

\(2 \sin \theta-\sqrt{2}=0\)

For Problems 37 through 42, use your knowledge of bearing, heading, and true course to sketch a diagram that will help you solve each problem. Resultant Force A heavy log is dragged across the ground by two horses pulling on ropes (Figure 9). The magnitudes of the tension forces in the direction of the ropes are 58 pounds and 73 pounds. If the angle between the ropes is \(26^{\circ}\), find the magnitude of the resultant force.

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=-54 \mathrm{i}, \mathrm{d}=20 \mathrm{i}\)

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(2 \mathbf{U}-3 \mathbf{V}\). $$\mathbf{U}=\langle 4,1\rangle, \mathbf{V}=\langle-5,2\rangle$$

Find the area of triangle \(A B C\) if \(A=56^{\circ}, B=71^{\circ}\), and \(c=21\) inches. a. \(150 \mathrm{in}^{2}\) b. \(200 \mathrm{in}^{2}\) c. 220 in \(^{2}\) d. \(240 \mathrm{in}^{2}\)

Find all radian solutions using exact values only. $$\sin x+\cos x=0$$

\(5 \tan \theta-3=0\)

Resultant Force Two trucks are trying to pull an auto out of the mud using chains. The magnitudes of the tension forces in the direction of the chains are 556 pounds and 832 pounds. If the angle between the chains is \(38.5^{\circ}\), find the magnitude of the resultant force.

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=13 \mathbf{j}, \mathrm{d}=44 \mathrm{i}\)

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(2 \mathbf{U}-3 \mathbf{V}\). $$\mathbf{U}=\langle-6,-3\rangle, \mathbf{V}=\langle-2,5\rangle$$

Find the semiperimeter of triangle \(A B C\) with \(a=17, b=41\), and \(c=28\). a. 86 b. 172 C. 43 d. 29

Find all radian solutions using exact values only. $$\sin x-\cos x=1$$

\(\sin \theta \cos \theta-2 \cos \theta=0\)

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=39 \mathbf{j}, \mathrm{d}=72 \mathrm{i}\)

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

If Heron's formula is used to find the area of triangle \(A B C\) having \(a=3\) meters, \(b=5\) meters, and \(c=6\) meters, which of the following shows the correct way to set up the formula? a. \(S=7 \sqrt{(10)(12)(13)}\) b. \(S=\sqrt{(4)(2)(1)}\) c. \(S=\sqrt{7(3)(5)(6)}\) d. \(S=\sqrt{7(4)(2)(1)}\)

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Use the law of sines to find \(C\) for triangle \(A B C\) if \(B=35^{\circ}, a=28\) feet, and \(b=19\) feet. a. \(23^{\circ}\) or \(87^{\circ}\) b. \(58^{\circ}\) c. \(58^{\circ}\) or \(87^{\circ}\) d. No triangle is possible

\(3 \sin \theta+2 \sin \theta \cos \theta=0\)

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(3 \mathbf{U}+2 \mathbf{V}\). $$\mathbf{U}=\mathbf{i}+4 \mathbf{j}, \mathbf{V}=7 \mathbf{i}-\mathbf{j}$$

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Given triangle \(A B C\) with \(B=35^{\circ}, a=28\) feet, and \(b=19\) feet, if the law of cosines is used to solve the triangle, what quadratic equation must first be solved? a. \(c^{2}-45.87 c+423=0\) b. \(c^{2}+45.87 c-1,145=0\) c. \(c^{2}-31.13 c-423=0\) d. \(c^{2}+31.3 c+1,145=0\)

\(2 \sin ^{2} \theta-3 \sin \theta=-1\)

Bike Frame Geometry Given \(B C=51 \mathrm{~cm}, B D=61 \mathrm{~cm}, C D=78 \mathrm{~cm}, \angle A B C=52^{\circ}\), and \(\angle A C B=65^{\circ}\), find the following. a. The length of the chainstay, \(A C\) b. \(\angle B C D\)

Work A package is pushed across a floor a distance of 75 feet by exerting a force of 41 pounds downward at an angle of \(20^{\circ}\) with the horizontal. How much work is done?

For each pair of vectors, find \(\mathbf{U}+\mathbf{V}, \mathbf{U}-\mathbf{V}\), and \(3 \mathbf{U}+2 \mathbf{V}\). $$\mathbf{U}=6 \mathbf{i}, \mathbf{V}=-8 \mathbf{j}$$

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. A plane headed due west is traveling with a constant speed of 214 miles per hour. The wind is blowing at a constant speed in the direction \(50.5^{\circ}\). If the ground speed of the plane is 145 miles per hour, what is its true course? a. \(295.5^{\circ}\) b. \(25.5^{\circ}\) or \(334.5^{\circ}\) c. \(290.2^{\circ}\) d. \(300.3^{\circ}\) or \(340.7^{\circ}\)

\(10 \cos ^{2} \theta+\cos \theta-3=0\)

Bike Frame Geometry Given \(B C=49 \mathrm{~cm}, B D=59 \mathrm{~cm}, \angle B A C=53^{\circ}, \angle A C B=69^{\circ}\), and \(\angle C B D=88^{\circ}\), find the following. a. The length of the chainstay, \(A C\) b. \(\angle B D C\)

Work A package is pushed across a floor a distance of 52 feet by exerting a force of 15 pounds downward at an angle of \(25^{\circ}\) with the horizontal. How much work is done?

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

Towing a Barge A barge is pulled by two tugboats. The first tugboat is traveling at a speed of 15 knots with heading \(130^{\circ}\), and the second tugboat is traveling at a speed of 16 knots with heading \(190^{\circ}\). Find the resulting speed and direction of the barge.

Work An automobile is pushed down a level street by exerting a force of 85 pounds at an angle of \(15^{\circ}\) with the horizontal (Figure 6). How much work is done in pushing the car 110 fect?

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

Pulling a Crate A large crate is pulled across the ice with two ropes. A force of 47 pounds is applied to the first rope in the direction \(80^{\circ}\), and a force of 55 pounds is applied to the second rope in the direction \(105^{\circ}\). What are the magnitude and direction of the resultant force acting on the crate?