Textbook Question
Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.
(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)
Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 8, Problem 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?
<IMAGE>
Verified step by step guidance1
Identify the forces acting on the monolith on the inclined causeway. The main force to consider is the component of the monolith's weight pulling it down the slope due to gravity.
Convert the weight of the monolith from tons to pounds (or another consistent unit) if necessary, since force is typically measured in pounds or newtons. For example, 1 ton = 2000 pounds, so multiply 60 tons by 2000 to get the weight in pounds.
Calculate the component of the weight acting parallel to the slope using the formula: \(F = W \times \sin(\theta)\), where \(W\) is the weight of the monolith and \(\theta\) is the slope angle (2.3°).
Use the sine of the slope angle to find the force pulling the monolith down the causeway. This force is the amount of force required to hold the monolith in place (i.e., to prevent it from sliding down).
Express the final force value as the force required to hold the monolith stationary on the slope, which balances the downhill component of the weight.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Was this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inclined Plane and Slope Angle
An inclined plane is a flat surface tilted at an angle to the horizontal. The slope angle determines how gravity affects objects on the plane, influencing the component of gravitational force acting along the slope. Understanding this helps calculate forces like the force needed to hold or move an object on the incline.
Recommended video:
04:46
Coterminal Angles
Decomposition of Forces on an Incline
When an object rests on an inclined plane, its weight can be split into two components: one perpendicular to the surface and one parallel to it. The parallel component causes the object to slide down, calculated as weight times the sine of the slope angle. This component is crucial for determining the force required to hold the object in place.
Recommended video:
3:22Example 2
Conversion of Units and Weight Calculation
Weight is the force due to gravity acting on a mass, calculated as mass times gravitational acceleration. Converting tons to pounds or kilograms and using the correct gravitational constant is essential for accurate force calculations. This ensures the force to hold the monolith is expressed in consistent units.
Recommended video:
06:11Introduction to the Unit Circle
Related Practice
Textbook Question
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
- 2u + 4v
Textbook Question
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?
<IMAGE>
1
views
Textbook Question
A force of 18.0 lb is required to hold a 60.0-lb stump grinder on an incline. What angle does the incline make with the horizontal?
Textbook Question
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?
Textbook Question
Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizontal.