Skip to main content
Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Lial 12th EditionCh. 7 - Applications of Trigonometry and Vectors문제 43
8장, 문제 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.


<IMAGE>

검증된 단계별 안내
1
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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
03:55
Position Vectors & Component Form

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.
추천 영상:
04:39
45-45-90 Triangles

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.
추천 영상:
04:42
Solve Trig Equations Using Identity Substitutions
관련 실천
교과서 질문

Given vectors u and v, find: v - 3u. 

u = 〈-1, 2〉, v = 〈3, 0〉

766
views
교과서 질문

Given vectors u and v, find: 2u + 3v.

u = 2i, v = i + j

866
views
교과서 질문

Given vectors u and v, find: 2u + 3v. 

u = 〈-1, 2〉, v = 〈3, 0〉

719
views
교과서 질문

Given vectors u and v, find: v - 3u. 

u = 2i, v = i + j

687
views
교과서 질문

Given vectors u and v, find: 2u. 

u = 〈-1, 2〉, v = 〈3, 0〉

765
views
교과서 질문

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?

796
views