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)
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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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### 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
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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