Chapter 4

Solve each problem. Suppose \(r\) varies directly with the square of \(m\) and inversely with \(s .\) If \(r=12\) when \(m=6\) and \(s=4,\) find \(r\) when \(m=4\) and \(s=10.\)

Solve each problem. Suppose \(p\) varies directly with the square of \(z\) and inversely with \(r .\) If \(p=\frac{32}{5}\) when \(z=4\) and \(r=10,\) find \(p\) when \(z=2\) and \(r=16\)

Solve each problem. If \(a\) varies directly with \(m\) and \(n^{2}\) and inversely with \(y^{3}\) and \(a=9\) when \(m=4, n=9,\) and \(y=3,\) find \(a\) if \(m=6, n=2,\) and \(y=5.\)

Solve each problem. If \(y\) varies directly with \(x\) and inversely with \(m^{2}\) and \(r^{2},\) and \(y=\frac{5}{3}\) when \(x=1, m=2,\) and \(r=3,\) find \(y\) if \(x=3\) \(m=1,\) and \(r=8.\)

Solve each problem. For \(k>0,\) if \(y\) varies directly with \(x,\) when \(x\) increases, \(y\) ______\(,\) and when \(x\) decreases, \(y\) ______.

For Exercises \(83-90\), the domains were determined in Exercises \(73-80 .\) Use a graph to (a) find the range, (b) give the largest open interval over which the function is increasing, (c) give the largest open interval over which the function is decreasing, and (d) solve the equation \(f(x)=0\) by observing the graph. $$f(x)=\sqrt[3]{8 x-24}$$

Solve each problem. For \(k>0,\) if \(y\) varies inversely with \(x,\) when \(x\) increases, \(y\) _______ and when \(x\) decreases, \(y\) _______.

For Exercises \(83-90\), the domains were determined in Exercises \(73-80 .\) Use a graph to (a) find the range, (b) give the largest open interval over which the function is increasing, (c) give the largest open interval over which the function is decreasing, and (d) solve the equation \(f(x)=0\) by observing the graph. $$f(x)=\sqrt[5]{x+32}$$

Assume that the constant of variation is positive. Let \(y\) be inversely proportional to \(x\). If \(x\) doubles, what happens to \(y ?\)

Assume that the constant of variation is positive. Let \(y\) vary inversely with the second power of \(x .\) If \(x\) doubles, what happens to \(y ?\)

Recall from Section 2.3 that if we are given the graph of \(y=f(x),\) we can obtain the graph of \(y=-f(x)\) by reflecting across the \(x\) -axis, and we can obtain the graph of \(y=f(-x)\) by reflecting across the \(y\) -axis. In Exercises \(87-90\), you are given the graph of a rational function \(y=f(x)\). Draw a sketch by hand of the graph of (a) \(y=-f(x)\) and (b) \(y=f(-x)\). (GRAPH CAN'T COPY).

Assume that the constant of variation is positive. Suppose \(y\) varies directly with the third power of \(x .\) If \(x\) triples, what happens to \(y ?\)

Concept Check Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt{9 x+27}$$

Each rational function has an oblique asymptote. Determine the equation of this asymptote. Then, use a graphing calculator to graph both the function and the asymptote in the window indicated. $$f(x)=\frac{2 x^{2}+3}{4-x} ;[-18.8,18.8] \text { by }[-50,25]$$

Assume that the constant of variation is positive. Suppose \(y\) is directly proportional to the second power of \(x .\) If \(x\) is halved, what happens to \(y ?\)

Concept Check Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt{16 x+16}$$

Each rational function has an oblique asymptote. Determine the equation of this asymptote. Then, use a graphing calculator to graph both the function and the asymptote in the window indicated. $$f(x)=\frac{x^{2}+9}{x+3} ;[-9.4,9.4] \text { by }[-25,25]$$

Solve each problem. The federal government has developed the body mass index (BMI) to determine ideal weights. A person's BMI is directly proportional to his or her weight in pounds and inversely proportional to the square of his or her height in inches. (A BMI of 19 to 25 corresponds to a healthy weight.) A 6-foot-tall person weighing 177 pounds has a BMI of \(24 .\) Find the BMI (to the nearest whole number) of a person whose weight is 130 pounds and whose height is 66 inches.

Concept Check Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt{4 x+16}+4$$

Each rational function has an oblique asymptote. Determine the equation of this asymptote. Then, use a graphing calculator to graph both the function and the asymptote in the window indicated. $$f(x)=\frac{x-x^{2}}{x+2} ;[-9.4,9.4] \text { by }[-15,25]$$

Solve each problem. Volume of a Gas Natural gas provides \(25 \%\) of U.S. energy. The volume of a gas varies inversely with the pressure and directly with the temperature. (Temperature must be measured in degrees Kelvin (K), a unit of measurement used in physics.) If a certain gas occupies a volume of 1.3 liters at \(300 \mathrm{K}\) and a pressure of 18 newtons per square centimeter, find the volume at \(340 \mathrm{K}\) and a pressure of 24 newtons per square centimeter.

Concept Check Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt{32-4 x}-3$$

Each rational function has an oblique asymptote. Determine the equation of this asymptote. Then, use a graphing calculator to graph both the function and the asymptote in the window indicated. $$f(x)=\frac{x^{2}+2 x}{1-2 x} ;[-4.7,4.7] \text { by }[-5,5]$$

Solve each problem. The electrical resistance \(R\) of a wire varies inversely with the square of its diameter \(d .\) If a 25 -foot wire with diameter 2 millimeters has resistance 0.5 ohm, find the resistance of a wire having the same length and diameter 3 millimeters.

Concept Check Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt[3]{27 x+54}-5$$

Becomes $$\begin{aligned}&f(x)=\frac{x^{5}+x^{4}+x^{2}+1}{x^{4}+1}\\\&f(x)=x+1+\frac{x^{2}-x}{x^{4}+1} \end{aligned}$$ after the numerator is divided by the denominator. (a) What is the equation of the oblique asymptote of the graph of the function? (b) For what \(x\) -value(s) does the graph of the function intersect its asymptote? (c) As \(x \rightarrow \infty,\) does the graph of the function approach its asymptote from above or below?

Solve each problem. According to Poiseuille's law, the resistance to flow of a blood vessel \(R\) is directly proportional to the length \(l\) and inversely proportional to the fourth power of the radius \(r .\) (Source: Hademenos, George J., "The Biophysics of Stroke," American Scientist, May-June 1997 .) If \(R=25\) when \(l=12\) and \(r=0.2,\) find \(R\) to the nearest hundredth as \(r\) increases to 0.3 while \(l\) is unchanged.

Concept Check Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt[3]{8 x-8}$$

Consider the rational function $$f(x)=\frac{x^{3}-4 x^{2}+x+6}{x^{2}+x-2}$$ Divide the numerator by the denominator and use the method of Example 3 to determine the equation of the oblique asymptote. Then, determine the coordinates of the point where the graph of \(f\) intersects its oblique asymptote. Use a calculator to support your answer.

Solve each problem. The weight of an object varies inversely with the square of its distance from the center of Earth. The radius of Earth is approximately 4000 miles. If a person weighs 160 pounds on Earth's surface, what would this individual weigh 8000 miles above the surface of Earth?

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x^{2}+y^{2}=1$$

Use long division of polynomials to show that for $$f(x)=\frac{x^{4}-5 x^{2}+4}{x^{2}+x-12}$$ if we divide the numerator by the denominator, then the quotient polynomial is \(x^{2}-x+8,\) and the remainder is \(-20 x+100 .\) Graph both \(f(x)\) and \(g(x)=x^{2}-x+8\) in the window \([-50,50]\) by \([0,1000] .\) Comment on the appearance of the two graphs. Explain how the graph of \(f\) approaches that of \(g\) as \(|x| \rightarrow \infty\).

Solve each problem. The brightness or intensity of starlight varies inversely with the square of its distance from Earth. The Hubble Telescope can see stars whose intensities are \(\frac{1}{50}\) of the faintest star now seen by ground-based telescopes. Determine how much farther the Hubble Telescope can see into space than ground-based telescopes.

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x^{2}+y^{2}=25$$

Suppose a friend tells you that the graph of $$f(x)=\frac{x^{2}-25}{x+5}$$ has a vertical asymptote with equation \(x=-5 .\) Is this correct? If not, describe the behavior of the graph at \(x=-5\).

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$(x-2)^{2}+(y+2)^{2}=4$$

Solve each problem. See Example 9. The strength \(S\) of a rectangular beam varies directly with its width \(W\) and the square of its thickness \(T,\) and inversely with its length \(L\). A beam that is 2 inches wide, 6 inches thick, and 96 inches long can support a load of 375 pounds. Determine how much a similar beam that is 3.5 inches wide, 8 inches thick, and 128 inches long can support.

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$(x+1)^{2}+y^{2}=9$$

Solve each problem involving rate of work. Linda and Tooney want to pick up the mess that their granddaughter, Kaylin, has made in her playroom. Tooney could do it in 15 minutes working alone. Linda, working alone, could clean it in 12 minutes. How long will it take them if they work together?

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x^{2}+(y-2)^{2}=16$$

Solve each problem involving rate of work. Johnny can groom Gary Bell's dogs in 6 hours, but it takes his business partner, "Mudcat," only 4 hours to groom the same dogs. How long will it take them to groom the dogs if they work together?

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$(x+3)^{2}+(y-1)^{2}=1$$

Solve each problem involving rate of work. Mrs. Schmulen is a high school mathematics teacher. She can grade a set of chapter tests in 5 hours working alone. If her student teacher Elwyn helps her, it will take 3 hours to grade the tests. How long would it take Elwyn to grade the tests if he worked alone?

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=y^{2}+1$$

Solve each problem involving rate of work. Tommy and Alicia are laying a tile floor. Working alone, Tommy can do the job in 20 hours. If the two of them work together, they can complete the job in 12 hours. How long would it take Alicia to lay the floor working alone?

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=y^{2}-3$$

Solve each problem involving rate of work. If a vat of solution can be filled by an inlet pipe in 5 hours and emptied by an outlet pipe in 10 hours, how long will it take to fill an empty vat if both pipes are open?

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$x=2 y^{2}$$

Solve each problem involving rate of work. A winery has a vat to hold Merlot. An inlet pipe can fill the vat in 18 hours, while an outlet pipe can empty it in 24 hours. How long will it take to fill an empty vat if both the outlet and the inlet pipes are open?

In Exercises \(97-108,\) graph by hand the equation of the circle or the parabola with a horizontal axis. $$c=-y^{2}$$