
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Question
![### 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.](https://content.bartleby.com/qna-images/question/df429993-eecf-4773-959c-7bbde76a8695/76c9128d-ef27-4f7a-b938-ef9ad93e4fce/xvml3eh_thumbnail.jpeg)
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.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution
Trending nowThis is a popular solution!
Step by stepSolved in 2 steps

Knowledge Booster
Similar questions
- What are the domain and range of y= A Domain: {xx = R} Range: {yly & R} B Domain (yly = R} Range: {xx ≥ 0, x = R} Select one: O a. A O b. B O c. C Od. D |6x² + 3x-3/² Solve Ix²+4x+121=2x+20 D c Domain: {xx ≤0, x = R} Range: {yly & R} D Domain: {xx = R} Range: {yly 20, y = R}arrow_forwardhelp!arrow_forwardFind the transition points. y = 15x³ + 360x² (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list) x = -8 Incorrect Find the interval(s) of increase. (Use symbolic notation and fractions where needed. Give your answer as interval(s) in the form (*, *). Use the symbol ∞ for infinity, U for combining intervals, and an appropriate type of parenthesis "(", ")", "[", "]" depending on whether the interval is open or closed. If the interval does not exist, enter Ø). x E (-∞0,-16) U (0,00) Find the interval(s) of decrease. (Use symbolic notation and fractions where needed. Give your answer as interval(s) in the form (*, *). Use the symbol ∞ for infinity, U for combining intervals, and an appropriate type of parenthesis "(", ")", "[", "]" depending on whether the interval is open or closed. If the interval does not exist, enter Ø). XE (-16,0) Find the interval(s) on which the function is concave up. (Use symbolic notation and fractions where…arrow_forward
- Which equation represents the set of all x-values that are exactly 42 units from 13?arrow_forward10. A rectangular prism has a volume, in cubic centimeter, represented byv(x)=2x2+5x2-14x-8 If the width is (x+4) cm, determine expressions for possibledimensions of the prism. please type the answer, not in writing thank youarrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Algebra and Trigonometry (6th Edition)AlgebraISBN:9780134463216Author:Robert F. BlitzerPublisher:PEARSONContemporary Abstract AlgebraAlgebraISBN:9781305657960Author:Joseph GallianPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
- Algebra And Trigonometry (11th Edition)AlgebraISBN:9780135163078Author:Michael SullivanPublisher:PEARSONIntroduction to Linear Algebra, Fifth EditionAlgebraISBN:9780980232776Author:Gilbert StrangPublisher:Wellesley-Cambridge PressCollege Algebra (Collegiate Math)AlgebraISBN:9780077836344Author:Julie Miller, Donna GerkenPublisher:McGraw-Hill Education

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

Contemporary Abstract Algebra
Algebra
ISBN:9781305657960
Author:Joseph Gallian
Publisher:Cengage Learning

Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning

Algebra And Trigonometry (11th Edition)
Algebra
ISBN:9780135163078
Author:Michael Sullivan
Publisher:PEARSON

Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:9780980232776
Author:Gilbert Strang
Publisher:Wellesley-Cambridge Press

College Algebra (Collegiate Math)
Algebra
ISBN:9780077836344
Author:Julie Miller, Donna Gerken
Publisher:McGraw-Hill Education