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Chapter 3: Problem 42

What does it mean if two quantities vary inversely?

Short Answer

Expert verified
When two quantities vary inversely, it means when one quantity increases, the other decreases in proportion, and vice versa. This is represented as \(xy = k\), where \(k\) is a constant. A typical real-life example is distributing a fixed amount of money among a group of people: as the number of people increases, the amount each one gets decreases, and vice versa.

Step by step solution

01

Definition of Inverse Variation

Inverse variation, or inverse proportionality, describes the relationship between two quantities such that when one quantity increases, the other decreases in proportion, and vice versa. It can be represented as \(xy = k\), where \(k\) is a constant of variation.
02

Explanation of the Concept

In this relationship, as one quantity doubles, the other one halves. This is because the product must always be equal to the constant \(k\). It's important to note that this relationship remains valid as long as neither of the quantities equates to zero.
03

Real-life Example

For instance, consider the scenario where you have a certain amount of money to distribute among people. As the number of people (quantity) increases, the amount of money each person will get (another quantity) decreases, here the total amount of money remains constant, hence showcasing variation inversely.

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Most popular questions from this chapter

Body-mass index, or BMI, takes both weight and height into account when assessing whether an individual is underweight or overweight. BMI varies directly as one's weight, in pounds, and inversely as the square of one's height, in inches. In adults, normal values for the BMI are between 20 and \(25,\) inclusive. Values below 20 indicate that an individual is underweight and values above 30 indicate that an individual is obese. A person who weighs 180 pounds and is 5 feet, or 60 inches, tall has a BMI of \(35.15 .\) What is the BMI, to the nearest tenth, for a 170 -pound person who is 5 feet 10 inches tall? Is this person overweight?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.

Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degreeof \(p\) and \(q\) are as small as possible. More than one correct finction may be possible. Graph your function using a graphing utility to verify that it has the required propertics \(f\) has vertical asymptotes given by \(x=-2\) and \(x=2\) a horizontal asymptote \(y=2, y\) -intercept at \(\frac{9}{2}, x\) -intercepts at \(-3\) and \(3,\) and \(y\) -axis symmetry.

In your own words, explain how to solve a variation problem.

Write an equation that expresses each relationship. Then solve the equation for \(y .\) \(x\) varies directly as \(z\) and inversely as the sum of \(y\) and \(w\).
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