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Chapter 1: Problem 56

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(y\) varies inversely as \(x .(y=3 \text { when } x=25 .)\)

Short Answer

Expert verified
The mathematical model representing the inverse variation is \(y = 75/x\). The constant of proportionality is \(75\).

Step by step solution

01

Understand the Concept of Inverse Variation

In mathematics, we say that one variable varies inversely as another when the product of the two variables is constant. This relationship can be represented by the equation \(y = k/x\), where \(k\) is the constant of variation or proportionality.
02

Use Given Condition to Determine Constant of Proportionality

The problem gives us a condition that when \(x = 25\), \(y = 3\). We can substitute these values into the inverse variation equation to solve for \(k\): \(3 = k / 25\)Solving this equation for \(k\) gives \(k = 3 * 25 = 75\).
03

Write the Final Mathematical Model

Therefore, the mathematical model that represents this inverse variation is \(y = 75/x\). This equation tells us that \(y\) and \(x\) are inversely proportional, and that the constant of proportionality is \(75\).

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. A function with a square root cannot have a domain that is the set of real numbers.

Sketch the graph of the function. $$f(x)=\left\\{\begin{array}{ll}1-(x-1)^{2}, & x \leq 2 \\\\\sqrt{x-2}, & x>2\end{array}\right.$$

Sketch the graph of the function. $$g(x)=\left\\{\begin{array}{ll}x+6, & x \leq-4 \\\\\frac{1}{2} x-4, & x>-4\end{array}\right.$$

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=\frac{5}{6}-\frac{2}{3} x$$

Find a mathematical model that represents the statement. (Determine the constant of proportionality.) \(z\) varies directly as the square of \(x\) and inversely as \(y\) \((z=6 \text { when } x=6 \text { and } y=4 .)\)
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