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Chapter 1: Problem 61

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

Short Answer

Expert verified
The mathematical model that represents the given condition is \(z = (2/3) \cdot x^2/y\)

Step by step solution

01

Understand the problem

The problem wants us to find a mathematical model in the form of an equation that represents the given condition. The condition is that \(z\) varies directly to the square of \(x\) and inversely to \(y\). We can represent this condition in the form \(z = k \cdot x^2/y\), where \(k\) is the constant of proportionality that we need to find.
02

Substitute the given values

Now we substitute the given values of \(x = 6\), \(y = 4\), and \(z = 6\) into the equation. This results in \(6 = k \cdot 6^2/4 = k \cdot 36/4\), which simplifies to \(6 = k \cdot 9\)
03

Solve for the constant of proportionality

Solving the equation \(6 = k \cdot 9\) for \(k\) gives us \(k = 6/9 = 2/3\). This is our constant of proportionality.
04

Formulate the final equation

Finally, we substitute \(k = 2/3\) back into the general equation to get the final equation \(z = (2/3) \cdot x^2/y\)

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Most popular questions from this chapter

(a) Write the linear function \(f\) such that it has the indicated function values and (b) Sketch the graph of the function. $$f\left(\frac{2}{3}\right)=-\frac{15}{2}, \quad f(-4)=-11$$

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$h(x)=\sqrt{x+2}+3$$

A balloon carrying a transmitter ascends vertically from a point 3000 feet from the receiving station. (a) Draw a diagram that gives a visual representation of the problem. Let \(h\) represent the height of the balloon and let \(d\) represent the distance between the balloon and the receiving station. (b) Write the height of the balloon as a function of \(d\) What is the domain of the function?

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=4+(1 / x)$$

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 440 vibrations per second. Find the frequency of a string that has 1.25 times as much tension and is 1.2 times as long.
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