Learn Before
Guitar String Frequency and Length: An Inverse Variation Application
Problem: The frequency of a guitar string varies inversely with its length. A 26-inch string has a frequency of 440 vibrations per second. (a) Write the equation of variation. (b) How many vibrations per second will there be if the string's length is reduced to 20 inches by placing a finger on a fret?
Part (a) — Find the equation.
Let = frequency and = length. Because varies inversely with , write the inverse variation formula:
Substitute the known values and :
Solve for the constant of variation by multiplying both sides by 26:
Substitute back into the formula:
Part (b) — Find when .
Substitute into the equation:
A 20-inch guitar string has a frequency of 572 vibrations per second. Shortening the string from 26 inches to 20 inches increases the frequency from 440 to 572, which is consistent with the inverse relationship: a shorter string vibrates more rapidly.
0
1
Tags
OpenStax
Elementary Algebra @ OpenStax
Ch.8 Rational Expressions and Equations - Elementary Algebra @ OpenStax
Algebra
Math
Prealgebra
Intermediate Algebra @ OpenStax
Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax
Related
Fuel Consumption and Car Weight: An Inverse Variation Application
Guitar String Frequency and Length: An Inverse Variation Application
A project manager at a construction firm knows that the time (y) required to pave a road varies inversely with the number of crew members (x) assigned. To find the specific equation for this relationship, the manager follows a standard four-step procedure. Arrange these steps in the correct order.
A logistics coordinator is using an inverse variation model to relate the average speed of a delivery truck (x) to the time required to complete a fixed route (y). According to the standard four-step procedure for solving inverse variation problems, which formula represents the correct starting point for this analysis?
A project manager is training a team to model workplace efficiency using inverse variation (for example, how increasing the number of staff reduces the time to complete a task). Match each component of the standard four-step procedure for solving inverse variation problems with its correct description or mathematical representation.
A maintenance supervisor uses an inverse variation model () to estimate how the size of a repair crew (x) affects the total hours needed to fix a machine (y). True or False: According to the standard four-step procedure, the constant of variation () is calculated by multiplying the number of crew members by the number of hours for a known repair.
Finalizing the Inverse Variation Model
An operations analyst is following the standard four-step procedure to find an inverse variation equation relating the number of technicians () to the repair time (). After substituting known values into the general formula , the analyst reaches the third step: solving for the constant of variation. To isolate the constant , the analyst must ________ both sides of the equation by the known value of .
Standardizing Inverse Variation Modeling for Logistics
Documenting the Standard Procedure for Inverse Variation Modeling
A data analyst is adapting a direct variation model into an inverse variation model to represent how increasing the number of server nodes reduces processing latency. According to the standard four-step procedure, which component of the process is the only one that is different when solving an inverse variation problem instead of a direct variation problem?
A technical support manager is using the standard four-step procedure to find an inverse variation equation that models customer wait times. After the manager has written the general formula (), what is the next action required by the procedure?
Solving an Inverse Variation Application: Time for Ice to Melt
Solving an Inverse Variation Application: Product Demand and Price
Learn After
In musical instrument design, the relationship between a string's frequency and its length is modeled by the inverse variation formula f = k / L. Match each variable in this formula to its corresponding physical or mathematical property.
A guitar technician is explaining to an apprentice that the frequency of a string varies inversely with its length. Which of the following best describes the behavior of this inverse relationship?
True or False: In a relationship where the frequency of a guitar string varies inversely with its length, shortening the string by placing a finger on a fret will result in a decrease in the vibration frequency.
A technician at a musical instrument factory needs to calculate how the frequency of a string changes when its length is modified. Arrange the following steps in the correct order to solve this inverse variation problem using the formula .
Algebraic Modeling of String Frequency
A technician at a musical instrument factory uses the inverse variation formula to model how the frequency of a string changes with its length. In this algebraic equation, the fixed value is specifically referred to as the ________ of variation.
Acoustic Engineering: String Length and Frequency
Acoustic Engineering Training: Variation Models
A quality control technician at a musical instrument factory is measuring several guitar strings to ensure they follow the expected inverse variation model. According to the principle of inverse variation (), which mathematical operation between a string's frequency () and its length () will always result in the constant of variation ()?
A musical instrument technician is testing a guitar string at various lengths by pressing it against different frets. According to the inverse variation model (), which of the following values remains unchanged for that specific string regardless of which fret is used?