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Chapter 3: Problem 55

Describe in words the variation shown by the given equation. \(z=\frac{k \sqrt{x}}{y^{2}}\)

Short Answer

Expert verified
The function \(z=\frac{k \sqrt{x}}{y^{2}}\) varies inversely with the square of 'y' and varies directly with the square root of 'x'.

Step by step solution

01

Identifying the type of variation when 鈥榶鈥� changes

Firstly, consider y independent of x. Looking at the function \(z=\frac{k}{y^{2}}\) which is known as the inverse square variation. Here, as 'y' increases by a certain factor, 'z' decreases by the square of that factor, assuming all other variables are kept constant. Conversely, as 'y' decreases by a certain factor, 'z' increases by the square of that factor.
02

Identifying the type of variation when 鈥榵鈥� changes

Secondly, consider x independent of y. Looking at the function \(z=k \sqrt{x}\), which is known as the direct square root variation. Here, when 'x' increases by a certain factor, 'z' increases by the square root of that factor, assuming all other variables are kept constant. Conversely, when 'x' decreases by a certain factor, 'z' decreases by the square root of that factor.
03

Conclusively Describing the Variation

In conclusion, the function \(z=\frac{k \sqrt{x}}{y^{2}}\) is simultaneously inversely proportional to the square of 'y' and directly proportional to the square root of 'x'. Consequently, when 'x' increases, 'z' increases, and when 'y' increases, 'z' decreases. Additionally, the factor of increase or decrease for 'z' depends on the square root for 'x' and the square of 'y'.

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Most popular questions from this chapter

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=4(x-3)(x+6)^{3}$$

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Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=3 x^{4}+4 x^{3}-7 x^{2}-2 x-3 $$

Use the four-step procedure for solving variation problems given on page 356 to solve. The volume of a gas varies directly as its temperature and inversely as its pressure. At a temperature of 100 Kelvin and a pressure of 15 kilograms per square meter, the gas occupies a volume of 20 cubic meters. Find the volume at a temperature of 150 Kelvin and a pressure of 30 kilograms per square meter.

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2(x-4)^{2}\left(x^{2}-25\right)$$
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