Textbook Question
Given vectors u and v, find: 2u.
u = 2i, v = i + j
Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
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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 42
A force of 30.0 lb is required to hold an 80.0-lb pressure washer on an incline. What angle does the incline make with the horizontal?
Verified step by step guidance1
Identify the forces acting on the pressure washer on the incline: the weight (80.0 lb) acting vertically downward and the force required to hold it (30.0 lb) acting along the incline.
Recognize that the component of the weight acting parallel to the incline is balanced by the holding force. This component can be expressed as \(W \sin(\theta)\), where \(W\) is the weight and \(\theta\) is the angle of the incline.
Set up the equation relating the holding force to the component of the weight along the incline: \(30.0 = 80.0 \sin(\theta)\).
Solve for \(\sin(\theta)\) by dividing both sides of the equation by 80.0: \(\sin(\theta) = \frac{30.0}{80.0}\).
Find the angle \(\theta\) by taking the inverse sine (arcsin) of the value obtained: \(\theta = \arcsin\left(\frac{30.0}{80.0}\right)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Resolving Forces on an Incline
When an object rests on an inclined plane, forces such as weight and applied force can be broken down into components parallel and perpendicular to the incline. Understanding how to resolve these forces helps analyze equilibrium and calculate unknown angles or forces.
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Example 2
Trigonometric Ratios (Sine, Cosine, Tangent)
Trigonometric ratios relate the angles of a right triangle to the ratios of its sides. In problems involving inclines, sine, cosine, and tangent functions are used to connect the incline angle with force components, enabling calculation of unknown angles or lengths.
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5:08Sine, Cosine, & Tangent of 30°, 45°, & 60°
Equilibrium of Forces
An object in equilibrium has net forces equal to zero. For the pressure washer held stationary on the incline, the applied force balances the component of its weight along the incline. Applying equilibrium conditions allows solving for the incline angle.
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04:3945-45-90 Triangles
Related Practice
Textbook Question
Given vectors u and v, find: 2u + 3v.
u = 2i, v = i + j
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Textbook Question
Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box.
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Textbook Question
Given vectors u and v, find: 2u + 3v.
u = 〈-1, 2〉, v = 〈3, 0〉
Textbook Question
Given vectors u and v, find: v - 3u.
u = 2i, v = i + j
Textbook Question
Given vectors u and v, find: 2u.
u = 〈-1, 2〉, v = 〈3, 0〉