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Find the polynomial representing the length of the fan belt shown below when a = 4, b = 5, and c = 3. The dimensions are in inches. Your answer will involve r. (Simplify your answer completely.) 3x + 4x + 4.5 x² - ax 1.5m x²+ cx + bx

Find the polynomial representing the length of the fan belt shown below when a = 4, b = 5, and c = 3. The dimensions are in inches. Your answer will involve r. (Simplify your answer completely.) 3x + 4x + 4.5 x² - ax 1.5m x²+ cx + bx

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
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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### Problem Statement #### Objective: Find the polynomial representing the length of the fan belt shown below when: - \( a = 4 \) - \( b = 5 \) - \( c = 3 \) The dimensions are in inches. Your answer will involve \( \pi \). Simplify your answer completely. #### Given Polynomial: \[ \left( 3x^2 + 4x + 4.5 \right) \text{ in } \] ### Diagram Description The fan belt system includes: 1. Three pulleys arranged in a triangular configuration. 2. The dimensions and segments of the belt path around the pulleys are given as polynomial expressions involving \( x \) and the constants \( a \), \( b \), and \( c \). #### Diagram Details: - **Top segment of the belt**: Labeled as \( x^2 - ax \) where \( a \) will be substituted with 4. - **Top right segment**: Labeled as \( 1.5 \pi \). This represents a half-turn around the top-right pulley. - **Bottom right segment**: Labeled as \( x^2 + bx \) where \( b \) will be substituted with 5. - **Bottom segment**: Labeled as \( 2 \pi \). This represents a complete turn around the bottom pulley. - **Bottom left segment**: Labeled as \( x^2 + cx \) where \( c \) will be substituted with 3. - **Top left segment**: Labeled as \( \pi \). This represents a half-turn around the top-left pulley. The belt path connects these segments: - From the top, goes around the top-right pulley, down to the bottom-right pulley, around the bottom pulley, then up to the bottom-left pulley, and finally back to the top-left pulley. ### Solution Steps: 1. Substitute \( a \), \( b \), and \( c \) in the polynomial expressions for each segment of the belt. 2. Sum all polynomial expressions and constant \(\pi\) segments to find the total length of the fan belt. 3. Simplify the resulting polynomial and constants to represent the total length in terms of \(x\) and \(\pi\).
Transcribed Image Text:### Problem Statement #### Objective: Find the polynomial representing the length of the fan belt shown below when: - \( a = 4 \) - \( b = 5 \) - \( c = 3 \) The dimensions are in inches. Your answer will involve \( \pi \). Simplify your answer completely. #### Given Polynomial: \[ \left( 3x^2 + 4x + 4.5 \right) \text{ in } \] ### Diagram Description The fan belt system includes: 1. Three pulleys arranged in a triangular configuration. 2. The dimensions and segments of the belt path around the pulleys are given as polynomial expressions involving \( x \) and the constants \( a \), \( b \), and \( c \). #### Diagram Details: - **Top segment of the belt**: Labeled as \( x^2 - ax \) where \( a \) will be substituted with 4. - **Top right segment**: Labeled as \( 1.5 \pi \). This represents a half-turn around the top-right pulley. - **Bottom right segment**: Labeled as \( x^2 + bx \) where \( b \) will be substituted with 5. - **Bottom segment**: Labeled as \( 2 \pi \). This represents a complete turn around the bottom pulley. - **Bottom left segment**: Labeled as \( x^2 + cx \) where \( c \) will be substituted with 3. - **Top left segment**: Labeled as \( \pi \). This represents a half-turn around the top-left pulley. The belt path connects these segments: - From the top, goes around the top-right pulley, down to the bottom-right pulley, around the bottom pulley, then up to the bottom-left pulley, and finally back to the top-left pulley. ### Solution Steps: 1. Substitute \( a \), \( b \), and \( c \) in the polynomial expressions for each segment of the belt. 2. Sum all polynomial expressions and constant \(\pi\) segments to find the total length of the fan belt. 3. Simplify the resulting polynomial and constants to represent the total length in terms of \(x\) and \(\pi\).
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