Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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### Understanding Polygon Diagonals

A polygon with \( n \) sides has \( D \) diagonals, where \( D \) is given by the function:

\[
D(n) = \frac{n(n-3)}{2}
\]

#### Problem

Find the number of sides \( n \) if \( 135 \leq D \leq 860 \).

### Solution Approach

1. **Identify the Equation**: Start with the formula for diagonals:
   \[ D = \frac{n(n-3)}{2} \]

2. **Set Up the Inequality**: Plug in the values for \( D \):
   \[ 135 \leq \frac{n(n-3)}{2} \leq 860 \]

3. **Solve for \( n \)**: Multiply through by 2 to eliminate the fraction:
   \[ 270 \leq n(n-3) \leq 1720 \]

4. **Determine the Range**: Solve the quadratic inequalities to find the range of possible \( n \).

#### Result

The number of sides is \( n \) within the interval:
\[
\{ n \mid [ \, \leq n \leq \, ] \}
\]

Replace the square brackets with the calculated values of \( n \).

### Summary

This example explores how to apply inequality and quadratic solving techniques to determine the potential number of sides of a polygon when a range of diagonal counts is provided.
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Transcribed Image Text:### Understanding Polygon Diagonals A polygon with \( n \) sides has \( D \) diagonals, where \( D \) is given by the function: \[ D(n) = \frac{n(n-3)}{2} \] #### Problem Find the number of sides \( n \) if \( 135 \leq D \leq 860 \). ### Solution Approach 1. **Identify the Equation**: Start with the formula for diagonals: \[ D = \frac{n(n-3)}{2} \] 2. **Set Up the Inequality**: Plug in the values for \( D \): \[ 135 \leq \frac{n(n-3)}{2} \leq 860 \] 3. **Solve for \( n \)**: Multiply through by 2 to eliminate the fraction: \[ 270 \leq n(n-3) \leq 1720 \] 4. **Determine the Range**: Solve the quadratic inequalities to find the range of possible \( n \). #### Result The number of sides is \( n \) within the interval: \[ \{ n \mid [ \, \leq n \leq \, ] \} \] Replace the square brackets with the calculated values of \( n \). ### Summary This example explores how to apply inequality and quadratic solving techniques to determine the potential number of sides of a polygon when a range of diagonal counts is provided.
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