The ropes, pulleys, and sandbags shown are part of a mechanical system used to raise and lower scenery for a stage play. For the scenery to be in the proper position, the following formula must apply. w2 = V w,2 + wa2 If w2 = 12.5 Ib and w3 = 7.5 lb, find w1. W1 Ib 37° 530 W3 W2
The ropes, pulleys, and sandbags shown are part of a mechanical system used to raise and lower scenery for a stage play. For the scenery to be in the proper position, the following formula must apply. w2 = V w,2 + wa2 If w2 = 12.5 Ib and w3 = 7.5 lb, find w1. W1 Ib 37° 530 W3 W2
Algebra and Trigonometry (6th Edition)
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ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Understanding the Use of Pulleys in a Mechanical System
The ropes, pulleys, and sandbags shown are part of a mechanical system used to raise and lower scenery for a stage play. For the scenery to be in the proper position, the following formula must apply:
\[ w_2 = \sqrt{w_1^2 + w_3^2} \]
Given:
- \( w_2 = 12.5 \) lb
- \( w_3 = 7.5 \) lb
Find \( w_1 \).
\[ w_1 = \]
**Diagram Explanation:**
The diagram shows a set of pulleys with three weights attached to the system:
- \( w_1 \) is hanging vertically from the left pulley.
- \( w_2 \) is hanging vertically from the center point of the system, where the ropes coming from the left and right pulleys meet.
- \( w_3 \) is hanging vertically from the right pulley.
Angles are marked at the point where ropes meet the center weight (\( w_2 \)):
- The angle between the rope on the left and the vertical axis is \( 37^\circ \).
- The angle between the rope on the right and the vertical axis is \( 53^\circ \).
This mechanical system illustrates how weights and angles interact to maintain balance and proper positioning of the stage scenery. By using the given formula, you can calculate the necessary weight \( w_1 \) to ensure the system is balanced and functional.
**Steps to Solve for \( w_1 \):**
1. Substitute \( w_2 \) and \( w_3 \) into the formula:
\[ 12.5 = \sqrt{w_1^2 + 7.5^2} \]
2. Square both sides to eliminate the square root:
\[ 12.5^2 = w_1^2 + 7.5^2 \]
\[ 156.25 = w_1^2 + 56.25 \]
3. Subtract \( 56.25 \) from both sides to solve for \( w_1^2 \):
\[ 156.25 - 56.25 = w_1^2 \]
\[ 100 = w_1^2 \]
4. Take the square root of both sides to find \( w_1 \):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8585c01c-e95f-4e9e-ab09-9be83264bdef%2Fd7d294f6-6a5a-4b5c-8634-16d57f811fe9%2Flz2nl1d_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding the Use of Pulleys in a Mechanical System
The ropes, pulleys, and sandbags shown are part of a mechanical system used to raise and lower scenery for a stage play. For the scenery to be in the proper position, the following formula must apply:
\[ w_2 = \sqrt{w_1^2 + w_3^2} \]
Given:
- \( w_2 = 12.5 \) lb
- \( w_3 = 7.5 \) lb
Find \( w_1 \).
\[ w_1 = \]
**Diagram Explanation:**
The diagram shows a set of pulleys with three weights attached to the system:
- \( w_1 \) is hanging vertically from the left pulley.
- \( w_2 \) is hanging vertically from the center point of the system, where the ropes coming from the left and right pulleys meet.
- \( w_3 \) is hanging vertically from the right pulley.
Angles are marked at the point where ropes meet the center weight (\( w_2 \)):
- The angle between the rope on the left and the vertical axis is \( 37^\circ \).
- The angle between the rope on the right and the vertical axis is \( 53^\circ \).
This mechanical system illustrates how weights and angles interact to maintain balance and proper positioning of the stage scenery. By using the given formula, you can calculate the necessary weight \( w_1 \) to ensure the system is balanced and functional.
**Steps to Solve for \( w_1 \):**
1. Substitute \( w_2 \) and \( w_3 \) into the formula:
\[ 12.5 = \sqrt{w_1^2 + 7.5^2} \]
2. Square both sides to eliminate the square root:
\[ 12.5^2 = w_1^2 + 7.5^2 \]
\[ 156.25 = w_1^2 + 56.25 \]
3. Subtract \( 56.25 \) from both sides to solve for \( w_1^2 \):
\[ 156.25 - 56.25 = w_1^2 \]
\[ 100 = w_1^2 \]
4. Take the square root of both sides to find \( w_1 \):
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