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Problem 7.30a
Use the figure to find each vector: u + v. Use vector notation as in Example 4.
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Problem 7.3a
In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?
a. two triangles
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Problem 7.30b
Use the figure to find each vector: u - v. Use vector notation as in Example 4.
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Problem 7.3b
In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?
b. exactly one triangle
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Problem 7.30c
Use the figure to find each vector: - v. Use vector notation as in Example 4.
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Problem 7.3c
In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?
c. no triangle
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Problem 7.31
Two tugboats are pulling a disabled speedboat into port with forces of 1240 lb and 1480 lb. The angle between these forces is 28.2°. Find the direction and magnitude of the equilibrant.
Problem 7.31
Solve each triangle. See Examples 2 and 3.
B = 74.8°, a = 8.92 in., c = 6.43 in.
Problem 7.31a
Use the figure to find each vector: u + v. Use vector notation as in Example 4.
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Problem 7.31b
Use the figure to find each vector: u - v. Use vector notation as in Example 4.
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Problem 7.31c
Use the figure to find each vector: - u. Use vector notation as in Example 4.
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Problem 7.32
Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is 24.5°. Find the direction and magnitude of the equilibrant.
Problem 7.33
Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle between the forces.
Problem 7.33
To find the distance AB across a river, a surveyor laid off a distance BC = 354 m on one side of the river. It is found that B = 112° 10' and C = 15° 20'. Find AB. See the figure.
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Problem 7.33
Solve each triangle. See Examples 2 and 3.
A = 112.8°, b = 6.28 m, c = 12.2 m
Problem 7.34
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
v - u
Problem 7.34
Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forces.
Problem 7.34
To determine the distance RS across a deep canyon, Rhonda lays off a distance TR = 582 yd. She then finds that T = 32° 50' and R = 102° 20'. Find RS. See the figure.
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Problem 7.35
A force of 176 lb makes an angle of 78° 50′ with a second force. The resultant of the two forces makes an angle of 41° 10′ with the first force. Find the magnitudes of the second force and of the resultant.
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Problem 7.35
A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km away is N 38.8° E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two observations of the lighthouse?
Problem 7.36
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
-5v
Problem 7.36
A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.
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Problem 7.36
Radio direction finders are placed at points A and B, which are 3.46 mi apart on an east-west line, with A west of B. From A the bearing of a certain radio transmitter is 47.7°, and from B the bearing is 302.5°. Find the distance of the transmitter from A.
Problem 7.37
The bearing of a lighthouse from a ship was found to be N 37° E. After the ship sailed 2.5 mi due south, the new bearing was N 25° E. Find the distance between the ship and the lighthouse at each location.
Problem 7.37
A force of 25 lb is required to hold an 80-lb crate on a hill. What angle does the hill make with the horizontal?
Problem 7.38
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
- 2u + 4v
Problem 7.38
Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizontal.
Problem 7.38
Standing on one bank of a river flowing north, Mark notices a tree on the opposite bank at a bearing of 115.45°. Lisa is on the same bank as Mark, but 428.3 m away. She notices that the bearing of the tree is 45.47°. The two banks are parallel. What is the distance across the river?
Problem 7.39
To build the pyramids in Egypt, it is believed that giant causeways were constructed to transport the building materials to the site. One such causeway is said to have been 3000 ft long, with a slope of about 2.3°. How much force would be required to hold a 60-ton monolith on this causeway?
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Problem 7.39
A balloonist is directly above a straight road 1.5 mi long that joins two villages. She finds that the town closer to her is at an angle of depression of 35°, and the farther town is at an angle of depression of 31°. How high above the ground is the balloon?
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