Chapter 5

A hunter is at a point on a riverbank. He wants to get to his cabin, located 3 miles north and 8 miles west. He can travel 5 mph along the river but only 2 mph on this very rocky land. How far upriver, to the nearest hundredth of a mile, should he go in order to reach the cabin in a minimum amount of time? (Hint: distance \(=\) rate \(\times\) time.) (Picture cant copy)

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

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{4}-5 x^{2}+4}{x^{4}-24 x^{2}+108}$$

For \(k>0,\) if \(y\) varies directly with \(x,\) when \(x\) increases, \(y\) ________. and when \(x\) decreases, \(y\) _________.

Graph each rational function by hand. Give the domain and range, and discuss symmetry. Give the equations of any asymptotes. $$f(x)=\frac{1}{x^{2}+2}$$

Cruise Ship Travel At noon, the cruise ship Celebration is 60 miles due south of the cruise ship Inspiration and is sailing north at a rate of 30 mph. If the Inspiration is sailing west at a rate of 20 mph, find the time at which the distance \(d\) between the ships is a minimum. What is this distance to the nearest hundredth of a mile? (Picture cant copy)

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 finction is increasing. (c) give the largest open interval over which the finction is decreasing, and (d) solve the equation \(f(x)=0\) by observing the graph. $$f(x)=\sqrt[3]{8 x-24}$$

For \(k>0,\) if \(y\) varies inversely with \(x,\) when \(x\) increases, \(y\) _____, and when \(x\) decreases, \(y\) _________.

Graph each rational function by hand. Give the domain and range, and discuss symmetry. Give the equations of any asymptotes. $$f(x)=\frac{1}{x^{2}+3}$$

"Mido" Simon is in his bass boat, the Katie, 3 miles from the nearest point on the shore. He wishes to reach his camp at Maggie Point, 6 miles farther down the shoreline. If Mido's motor is disabled, and he can row his boat at a rate of 4 mph and walk at a rate of 5 mph, find the least amount of time, to the nearest hundredth of an hour, that he will need to reach the camp. (Picture cant copy)

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

Graph each rational function by hand. Give the domain and range, and discuss symmetry. Give the equations of any asymptotes. $$f(x)=\frac{-x^{2}}{x^{2}+1}$$

Incorporate many concepts from earlier work with the method of solving equations involving roots. Work them in order. Consider the equation $$\sqrt[3]{4 x-4}=\sqrt{x+1}$$ Rewrite the equation, using rational exponents.

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 finction is increasing. (c) give the largest open interval over which the finction is decreasing, and (d) solve the equation \(f(x)=0\) by observing the graph. $$f(x)=\sqrt{49-x^{2}}$$

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

Graph each rational function by hand. Give the domain and range, and discuss symmetry. Give the equations of any asymptotes. $$f(x)=\frac{-2 x^{2}}{x^{2}+2}$$

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 finction is increasing. (c) give the largest open interval over which the finction is decreasing, and (d) solve the equation \(f(x)=0\) by observing the graph. $$f(x)=\sqrt{81-x^{2}}$$

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

Graph each rational function by hand. Give the domain and range, and discuss symmetry. Give the equations of any asymptotes. $$f(x)=\frac{2 x^{2}}{x^{4}+1}$$

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. $$y=\sqrt{9 x+27}$$

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 \(y ?\)

Graph each rational function by hand. Give the domain and range, and discuss symmetry. Give the equations of any asymptotes. $$f(x)=\frac{-2 x^{2}}{x^{4}+1}$$

Solve each problem. Body Mass Index The federal government has devel. oped 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.

Use a calculator to graph rational function in the window indicated. Then (a) give the \(x\) - and y-intercepts, (b) explain why there are no vertical asymptotes, (c) give the equation of the oblique asymptote, and (d) give the domain and range. $$f(x)=\frac{3 x^{3}+2 x^{2}-12 x-8}{x^{2}+x+4} ;[-6.6,6.6] \text { by }[-4.1,4.1]$$

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. $$y=\sqrt{4 x+16}+4$$

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 kelvins (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.

Use a calculator to graph rational function in the window indicated. Then (a) give the \(x\) - and y-intercepts, (b) explain why there are no vertical asymptotes, (c) give the equation of the oblique asymptote, and (d) give the domain and range. $$f(x)=\frac{4 x^{3}+8 x^{2}-36 x-72}{2 x^{2}-x+6} ;[-5,5] \text { by }[-20,15]$$

Incorporate many concepts from earlier work with the method of solving equations involving roots. Work them in order. Consider the equation $$\sqrt[3]{4 x-4}=\sqrt{x+1}$$ Use synthetic division to show that 3 is a zero of the polynomial $$P(x)=x^{3}-13 x^{2}+35 x-15$$

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. $$y=\sqrt{32-4 x}-3$$

Solve each problem. Electrical Resistance 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.

Use a calculator to graph rational function in the window indicated. Then (a) give the \(x\) - and y-intercepts, (b) explain why there are no vertical asymptotes, (c) give the equation of the oblique asymptote, and (d) give the domain and range. $$f(x)=\frac{x^{3}+4 x^{2}-x-4}{-2 x^{2}-2 x-4} ;[-6.6,6.6] \text { by }[-4.1,4.1]$$

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. $$y=\sqrt[3]{27 x+54}-5$$

Use a calculator to graph rational function in the window indicated. Then (a) give the \(x\) - and y-intercepts, (b) explain why there are no vertical asymptotes, (c) give the equation of the oblique asymptote, and (d) give the domain and range. $$f(x)=\frac{-x^{3}-7 x^{2}+16 x+112}{x^{2}+x+28} ;[-15,10] \text { by }[-5,15]$$

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. $$y=\sqrt[3]{8 x-8}$$

Solve each problem. Gravity 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?

Incorporate many concepts from earlier work with the method of solving equations involving roots. Work them in order. Consider the equation $$\sqrt[3]{4 x-4}=\sqrt{x+1}$$ What are the three proposed solutions of the original equation, $$\sqrt[3]{4 x-4}=\sqrt{x+1} ?$$

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x^{2}+y^{2}=1$$

Solve each problem. Hubble Telescope 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.

Graph by hand the equation of the circle or the parabola with a horizontal axis. $$x^{2}+y^{2}=25$$

Solve each problem. Volume of a Cylinder The volume of a right circular cylinder is jointly proportional to the square of the radius of the circular base and to the height. If the volume is 300 cubic centimeters when the height is 10.62 centimeters and the radius is 3 centimeters, approximate the volume of a cylinder with radius 4 centimeters and height 15.92 centimeters. (image can't copy)

Incorporate many concepts from earlier work with the method of solving equations involving roots. Work them in order. Consider the equation $$\sqrt[3]{4 x-4}=\sqrt{x+1}$$ Use both an analytic method and your calculator to solve the original equation.

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. Strength of a Beam See Example \(11 .\) 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.

Incorporate many concepts from earlier work with the method of solving equations involving roots. Work them in order. Consider the equation $$\sqrt[3]{4 x-4}=\sqrt{x+1}$$ Write an explanation of how the solutions of the equation in Exercise 92 relate to the solutions of the original equation. Discuss any extraneous solutions that may be involved.

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. Two grandparents want to pick up the mess that their granddaughter has made in her playroom. One can do it in 15 minutes working alone. The other, working alone, can clean it in 12 minutes. How long will it take them if they work together?

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. One person can groom a dog in 6 hours, but it takes his business partner only 4 hours to groom the same dog. How long will it take them to groom the dog if they work together?

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. A high school mathematics teacher can grade a set of chapter tests in 5 hours working alone. If her student teacher helps her, it will take them 3 hours to grade the tests. How long would it take the student teacher to grade the tests if he worked alone?