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Chapter 7: Problem 46

What does it mean if two quantities vary inversely?

Short Answer

Expert verified
Two quantities are said to vary inversely if an increase in one quantity results in a decrease in the other quantity and vice versa. Their product remains constant. For example, the relationship between speed and time taken to travel a certain distance is an inverse variation.

Step by step solution

01

Define Inverse Variation

Inverse variation or inverse proportionality is a relationship between two variables, where one variable increases as the other decreases. Mathematically, it can be described as \(xy = k\), where \(x\) and \(y\) are the two variables and \(k\) is a constant.
02

Provide an example

An example of inverse variation is the relationship between speed and time taken to travel a certain distance. If you increase the speed (variable \(x\)), the time taken (variable \(y\)) to travel a certain distance decreases. Likewise, if the speed decreases, the time taken to travel the same distance will increase. Thus, the product of the speed and time remains constant (\(xy = k\)), showing that they vary inversely.

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Most popular questions from this chapter

Two formulas that approximate the dosage of a drug prescribed for children are $$ \begin{aligned} \text { Young's rule: } & C=\frac{D A}{A+12} \\ \text { and Cowling's rule: } & C=\frac{D(A+1)}{24} \end{aligned} $$ In each formula, \(A=\) the child's age, in years, \(D=\) an adult dosage, and \(C=\) the proper child's dosage. The formulas apply for ages 2 through \(13,\) inclusive. Use the formulas to solve Exercises \(93-96\) Use Young's rule to find the difference in a child's dosage for a 10 -year-old child and a 3 -year-old child. Express the answer as a single rational expression in terms of \(D .\) Then describe what your answer means in terms of the variables in the model.

In Exercises \(87-90\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. All real numbers satisfy the equation \(\frac{3}{x}-\frac{1}{x}=\frac{2}{x}\)

Solve: $$8(2-x)=-5 x$$

Add or subtract as indicated. Simplify the result, if possible. $$\frac{y}{y^{2}-1}+\frac{2 y}{y-y^{2}}$$

In Exercises \(94-96,\) use a graphing utility to solve each rational equation. Graph each side of the equation in the given viewing rectangle. The first coordinate of each point of intersection is a solution. Check by direct substitution. $$\begin{aligned} &\frac{50}{x}=2 x\\\ &[-10,10,1] \text { by }[-20,20,2] \end{aligned}$$
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