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 x. (Simplify your answer completely.) in 2-ax 1.5 x² + cx x² + 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 x. (Simplify your answer completely.) in 2-ax 1.5 x² + cx x² + bx
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:
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 \( \pi \). (Simplify your answer completely.)
#### Diagram Explanation:
The provided diagram shows a fan belt looping around three pulleys. The length segments are marked as follows:
- The segment on top is denoted as \( x^2 - ax \) inches
- The segment going to the right is \( x^2 + bx \) inches
- The segment going to the left is \( x^2 + cx \) inches
- The top circular segment is \( \pi \) radians
- The bottom circular segment is \( 2 \pi \) radians
- The middle circular segment is \( 1.5 \pi \) radians
### Solution:
1. **Identify Each Segment Length**:
- Top segment: \( x^2 - ax = x^2 - 4x \) inches (since \( a = 4 \))
- Right segment: \( x^2 + bx = x^2 + 5x \) inches (since \( b = 5 \))
- Left segment: \( x^2 + cx = x^2 + 3x \) inches (since \( c = 3 \))
2. **Sum Up Each Segment**:
- Linear segments: \( (x^2 - 4x) + (x^2 + 5x) + (x^2 + 3x) \)
- Combine like terms:
\[
(x^2 - 4x) + (x^2 + 5x) + (x^2 + 3x) = x^2 + x^2 + x^2 + (-4x) + 5x + 3x = 3x^2 + 4x
\]
3. **Sum Up Circular Segments**:
- \( \pi + 2\pi + 1.5\pi = 4.5\pi \)
4. **Total Belt Length**:
\[
\text{Total Length} = \text{Linear Length} + \text{Circular Length}
\]
\[
\text{Total Length} = 3x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c02ecae-9f3c-4011-90ce-878cb1c02c8e%2F4d0f3dfe-8327-4cc1-9a21-082a8094a482%2Fxcujcsm_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
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 \( \pi \). (Simplify your answer completely.)
#### Diagram Explanation:
The provided diagram shows a fan belt looping around three pulleys. The length segments are marked as follows:
- The segment on top is denoted as \( x^2 - ax \) inches
- The segment going to the right is \( x^2 + bx \) inches
- The segment going to the left is \( x^2 + cx \) inches
- The top circular segment is \( \pi \) radians
- The bottom circular segment is \( 2 \pi \) radians
- The middle circular segment is \( 1.5 \pi \) radians
### Solution:
1. **Identify Each Segment Length**:
- Top segment: \( x^2 - ax = x^2 - 4x \) inches (since \( a = 4 \))
- Right segment: \( x^2 + bx = x^2 + 5x \) inches (since \( b = 5 \))
- Left segment: \( x^2 + cx = x^2 + 3x \) inches (since \( c = 3 \))
2. **Sum Up Each Segment**:
- Linear segments: \( (x^2 - 4x) + (x^2 + 5x) + (x^2 + 3x) \)
- Combine like terms:
\[
(x^2 - 4x) + (x^2 + 5x) + (x^2 + 3x) = x^2 + x^2 + x^2 + (-4x) + 5x + 3x = 3x^2 + 4x
\]
3. **Sum Up Circular Segments**:
- \( \pi + 2\pi + 1.5\pi = 4.5\pi \)
4. **Total Belt Length**:
\[
\text{Total Length} = \text{Linear Length} + \text{Circular Length}
\]
\[
\text{Total Length} = 3x
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