The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string. The middle A string has a frequency of 430 vibrations per second. Find the frequency of a string that has 1.25 times as much tension and is 1.4 times as long. 373.4 vibrations / sec 383.4 vibrations / sec 343.4 vibrations / sec 363.4 vibrations / sec 353.4 vibrations / sec
The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string. The middle A string has a frequency of 430 vibrations per second. Find the frequency of a string that has 1.25 times as much tension and is 1.4 times as long. 373.4 vibrations / sec 383.4 vibrations / sec 343.4 vibrations / sec 363.4 vibrations / sec 353.4 vibrations / sec
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,...
Related questions
Question
![**Problem Statement:**
The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string. The middle A string has a frequency of 430 vibrations per second. Find the frequency of a string that has 1.25 times as much tension and is 1.4 times as long.
**Multiple Choice Answers:**
- ○ 373.4 vibrations / sec
- ○ 383.4 vibrations / sec
- ● 343.4 vibrations / sec
- ○ 363.4 vibrations / sec
- ○ 353.4 vibrations / sec
**Explanation:**
To solve this problem, we use the relationship between frequency \( f \), tension \( T \), and length \( L \) of the string:
\[
f \propto \frac{\sqrt{T}}{L}
\]
Given:
- Original frequency \( f_1 = 430 \) vibrations/sec
- New tension \( T_2 = 1.25T \)
- New length \( L_2 = 1.4L \)
New frequency \( f_2 \):
\[
f_2 = f_1 \times \sqrt{\frac{T_2}{T}} \times \frac{L}{L_2}
\]
\[
f_2 = 430 \times \sqrt{\frac{1.25T}{T}} \times \frac{1}{1.4}
\]
\[
f_2 = 430 \times \sqrt{1.25} \times \frac{1}{1.4}
\]
\[
f_2 = 430 \times 1.118 \times 0.714
\]
\[
f_2 = 343.4 \text{ vibrations/sec}
\]
The correct answer is 343.4 vibrations/sec (option 3).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F556d7fe8-ec0a-4648-8d64-f7fe013c7dbe%2F52133faf-c741-4281-99b7-e31131f15881%2Fsrujqxf_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The frequency of vibrations of a piano string varies directly as the square root of the tension on the string and inversely as the length of the string. The middle A string has a frequency of 430 vibrations per second. Find the frequency of a string that has 1.25 times as much tension and is 1.4 times as long.
**Multiple Choice Answers:**
- ○ 373.4 vibrations / sec
- ○ 383.4 vibrations / sec
- ● 343.4 vibrations / sec
- ○ 363.4 vibrations / sec
- ○ 353.4 vibrations / sec
**Explanation:**
To solve this problem, we use the relationship between frequency \( f \), tension \( T \), and length \( L \) of the string:
\[
f \propto \frac{\sqrt{T}}{L}
\]
Given:
- Original frequency \( f_1 = 430 \) vibrations/sec
- New tension \( T_2 = 1.25T \)
- New length \( L_2 = 1.4L \)
New frequency \( f_2 \):
\[
f_2 = f_1 \times \sqrt{\frac{T_2}{T}} \times \frac{L}{L_2}
\]
\[
f_2 = 430 \times \sqrt{\frac{1.25T}{T}} \times \frac{1}{1.4}
\]
\[
f_2 = 430 \times \sqrt{1.25} \times \frac{1}{1.4}
\]
\[
f_2 = 430 \times 1.118 \times 0.714
\]
\[
f_2 = 343.4 \text{ vibrations/sec}
\]
The correct answer is 343.4 vibrations/sec (option 3).
Transcribed Image Text:**Problem Statement:**
Find a mathematical model representing the statement. (Determine the constant of proportionality.)
**Given:**
\( z \) varies directly as the square of \( x \) and inversely as \( y \).
- When \( z = 36 \), \( x = 9 \), and \( y = 3 \).
**Options:**
1. \( z = \frac{4x^2}{3y} \)
2. \( z = -\frac{3x^2}{4y} \)
3. \( z = \frac{3x^2}{4y} \)
4. \( z = -\frac{4x^2}{3y} \)
5. \( z = \frac{4x}{3y} \)
**Explanation:**
Each option presents a mathematical equation involving \( x \), \( y \), and \( z \) to determine the correct expression by applying the given condition where \( z \), \( x \), and \( y \) have specific values. The task is to identify the equation that satisfies the proportional relationship specified.
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 now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
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 (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