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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblem 43
Chapter 8, Problem 43

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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Identify the forces exerted by the two people and their respective angles with the horizontal. Let the first force be \(F_1 = 150\) lb at an angle \(\theta_1 = 62.4^\circ\), and the second force be \(F_2 = 114\) lb at an angle \(\theta_2 = 54.9^\circ\).
Resolve each force into its horizontal and vertical components using trigonometric functions: For each force, the horizontal component is \(F \cos(\theta)\) and the vertical component is \(F \sin(\theta)\). So, calculate \(F_{1x} = 150 \cos(62.4^\circ)\), \(F_{1y} = 150 \sin(62.4^\circ)\), \(F_{2x} = 114 \cos(54.9^\circ)\), and \(F_{2y} = 114 \sin(54.9^\circ)\).
Since the box is being carried steadily (in equilibrium), the net horizontal force must be zero. Use this to check or confirm the forces balance horizontally: \(F_{1x} + F_{2x} = 0\) or consider if any friction or other forces are acting horizontally.
The weight of the box corresponds to the total vertical force supporting it. Sum the vertical components of the two forces to find the total upward force: \(W = F_{1y} + F_{2y}\), where \(W\) is the weight of the box.
Express the weight of the box as the sum of the vertical components calculated, which gives the magnitude of the weight force acting downward, balanced by the upward forces from the two people.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vector Components of Forces

Forces acting at angles can be broken down into horizontal and vertical components using trigonometric functions. The horizontal component is found by multiplying the force by the cosine of the angle, and the vertical component by the sine. This decomposition allows for easier analysis of combined forces.
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Resultant Force and Equilibrium

When multiple forces act on an object, their vector sum (resultant force) determines the net effect. If the box is stationary, the vertical components of the forces must balance the weight, meaning the sum of vertical forces equals the weight, and horizontal forces cancel out for equilibrium.
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Using Trigonometry to Solve for Unknowns

Trigonometric relationships enable solving for unknown forces or weights by relating angles and force components. By applying sine and cosine functions to given angles and forces, one can set up equations to find the weight of the box from the vertical force components.
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Related Practice
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Textbook Question

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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

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u = 〈-1, 2〉, v = 〈3, 0〉

Textbook Question

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?