Chapter 5

Solve the equation in part (a) graphically, expressing solutions to the nearest hundredth. Then use the graph to solve the associated inequalities in parts (b) and (c), expressing endpoints to the nearest hundredth. (a) \(\frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}=0\) (b) \(\frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}>0\) (c) \(\frac{\sqrt{2} x+5}{x^{3}-\sqrt{3}}<0\)

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}-1}{x^{2}-x}$$

Heavier birds have larger wings with more surface area than do lighter birds. For some species of birds, this relationship is given by $$S(x)=0.2 x^{2 / 3}$$ where \(x\) is the weight of the bird in kilograms and \(S\) is the surface area of the wings in square meters. (Source: Pennycuick, \(C\)., Newton Rules Biology. Oxford University Press.) Approximate \(S(0.5)\) and interpret the result.

Solve each equation involving "nested" radicals for all real solutions analytically. Support your solutions with a graph. $$\sqrt{\sqrt{x}}=x$$

Solve the equation in part (a) graphically, expressing solutions to the nearest hundredth. Then use the graph to solve the associated inequalities in parts (b) and (c), expressing endpoints to the nearest hundredth. (a) \(\frac{\sqrt[3]{7} x^{3}-1}{x^{2}+2}=0\) (b) \(\frac{\sqrt[3]{7} x^{3}-1}{x^{2}+2}>0\) (c) \(\frac{\sqrt[3]{7} x^{3}-1}{x^{2}+2}<0\)

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{9-x^{2}}{3 x-x^{2}}$$

Explain why determining the domain of a function of the form $$f(x)=\sqrt[n]{a x+b}$$ requires two different considerations, depending upon the parity of \(n .\)

Solve each equation involving "nested" radicals for all real solutions analytically. Support your solutions with a graph. $$\sqrt[3]{\sqrt[3]{x}}=x$$

Insect Population Suppose that an insect population in millions is modeled by $$ f(x)=\frac{10 x+1}{x+1} $$ where \(x \geq 0\) is in months. (a) Graph \(f\) in the window \([0,14]\) by \([0,14] .\) Find the equation of the horizontal asymptote. (b) Determine the initial insect population. (c) What happens to the population after several months? (d) Interpret the horizontal asymptote.

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

Solve each equation involving "nested" radicals for all real solutions analytically. Support your solutions with a graph. $$\sqrt{\sqrt{28 x+8}}=\sqrt{3 x+2}$$

Fish Population Suppose that the population of a species of fish in thousands is modeled by $$ f(x)=\frac{x+10}{0.5 x^{2}+1} $$ where \(x \geq 0\) is in years. (a) Graph \(f\) in the window \([0,12]\) by \([0,12] .\) What is the equation of the horizontal asymptote? (b) Determine the initial population. (c) What happens to the population after many years? (d) Interpret the horizontal asymptote.

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}-2 x+1}{2 x^{2}-2}$$

Determine the domain of each function. $$f(x)=\sqrt{9 x+18}$$

Solve each equation involving "nested" radicals for all real solutions analytically. Support your solutions with a graph. $$\sqrt{\sqrt{2 x+10}}=\sqrt{2 x-2}$$

Time Spent in Line Suppose the average number of vehicles arriving at the main gate of an amusement park is equal to 10 per minute, while the average number of vehicles being admitted through the gate per minute is equal to \(x .\) Then the average waiting time in minutes for each vehicle at the gate is given by $$ f(x)=\frac{x-5}{x^{2}-10 x} $$ where \(x>10\). (Source: Mannering, F. and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, 2d. ed., John Wiley and Sons.) (a) Estimate the admittance rate \(x\) that results in an average wait of 15 seconds. (b) If one attendant can serve 5 vehicles per minute, how many attendants are needed to keep the average wait to 15 seconds or less?

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}+2 x}{2 x-1}$$

Determine the domain of each function. $$f(x)=-\sqrt[4]{6-x}$$

Solve each equation involving "nested" radicals for all real solutions analytically. Support your solutions with a graph. $$\sqrt[3]{\sqrt{32 x}}=\sqrt[3]{x+6}$$

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

Determine the domain of each function. $$f(x)=-\sqrt[4]{2-0.5 x}$$

Solve each equation involving "nested" radicals for all real solutions analytically. Support your solutions with a graph. $$\sqrt[3]{\sqrt{x+63}}=\sqrt[3]{2 x+6}$$

Construction Find possible dimensions for a closed box with volume 196 cubic inches, surface area 280 square inches, and length that is twice the width.

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}-9}{x+3}$$

Determine the domain of each function. $$f(x)=\sqrt[3]{8 x-24}$$

The velocity \(v\) of a meteorite approaching Earth is given by $$v=\frac{k}{\sqrt{d}}$$ measured in kilometers per second, where \(d\) is its distance from the center of Earth and \(k\) is a constant. If \(k=350\) what is the velocity of a meteorite that is 6000 kilometers away from the center of Earth? Round to the nearest tenth.

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}-16}{x+4}$$

Determine the domain of each function. $$f(x)=\sqrt[5]{x+32}$$

The illumination \(I\) in foot-candles produced by a light source is related to the distance \(d\) in feet from the light source by the equation $$d=\sqrt{\frac{k}{I}},$$ where \(k\) is a constant. If \(k=400,\) how far from the source will the illumination be 14 foot-candles? Round to the nearest hundredth of a foot.

Train Curves When curves are designed for trains, sometimes the outer rail is elevated or banked so that a locomotive can safely negotiate the curve at a higher speed. Suppose a circular curve is being designed for a speed of 60 mph. The rational function \(f(x)=\frac{2540}{x}\) computes the elevation \(y\) in inches of the outer track for a curve with a radius of \(x\) feet, where \(y=f(x) .\) (Image can't copy) (a) Evaluate \(f(400)\) and interpret its meaning. (b) Graph \(f\) in the window \([0,600]\) by \([0,50] .\) Discuss how the elevation of the outer rail changes with the radius \(x\) (c) Interpret the horizontal asymptote. (d) What radius is associated with an elevation of 12.7 inches?

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

Determine the domain of each function. $$f(x)=\sqrt{49-x^{2}}$$

Period of a Pendulum The period \(P\) of a pendulum in seconds depends on its length \(L\) in feet and is given by $$P=2 \pi \sqrt{\frac{L}{32}}$$ If the length of a pendulum is 5 feet, what is its period? Round to the nearest tenth.

Recycling A cost-benefit function \(C\) computes the cost in millions of dollars of implementing a city recycling project when \(x\) percent of the citizens participate, where $$ C(x)=\frac{1.2 x}{100-x} $$ (a) Graph \(C\) in the window \([0,100]\) by \([0,10]\). Interpret the graph as \(x\) approaches 100 (b) If \(75 \%\) participation is expected, determine the cost for the city. (c) The city plans to spend \(\$ 5\) million on this recycling project. Estimate graphically the percentage of participation that they are expecting. (d) Solve part (c) analytically.

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}-5}{x-3}$$

Determine the domain of each function. $$f(x)=\sqrt{81-x^{2}}$$

A formula for calculating the distance \(d\) one can see from an airplane to the horizon on a clear day is $$d=1.22 \sqrt{x}$$ where \(x\) is the altitude of the plane in feet and \(d\) is given in miles. How far can one see to the horizon in a plane flying at each altitude? Give answers to the nearest mile. (a) \(15,000\) feet (b) \(20,000\) feet

Braking Distance The grade \(x\) of a hill is a measure of its steepness. For example, if a road rises 10 feet for every 100 feet of horizontal distance, then it has an uphill grade of \(x=\frac{10}{100},\) or \(10 \% .\) The braking (or stopping) distance \(D\) for a car traveling at 50 mph on a wet, uphill grade is given by $$ D(x)=\frac{2500}{30(0.3+x)} $$ (a) Evaluate \(D(0.05)\) and interpret the result. (b) Describe what happens to braking distance as the hill becomes steeper. (c) Estimate the grade associated with a braking distance of 220 feet.

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}-1}{x^{2}-4 x+3}$$

Determine the domain of each function. $$f(x)=\sqrt{x^{3}-x}$$

Braking Distance See Exercise \(81 .\) If a car is traveling 50 mph downhill, then its braking distance on wet pavement is given by $$ D(x)=\frac{2500}{30(0.3+x)} $$ where \(x<0\) for a downhill grade. (a) Evaluate \(D(-0.1)\) and interpret the result. (b) What happens to braking distance as the downhill grade becomes steeper? (c) The graph of \(D\) has a vertical asymptote at \(x=-0.3\) Give the physical significance of this asymptote. (d) Estimate the grade associated with a braking distance of 350 feet.

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}-4}{x^{2}+3 x+2}$$

Explain why the domain of \(f(x)=\sqrt{x^{2}+1}\) is \((-\infty, \infty)\)

A research biologist has shown that the number \(S\) of different plant species on a Galápagos Island is related to the area \(\mathscr{A}\) of the island by $$S=28.6 \sqrt[3]{\mathscr{A}}$$ Find \(S\) for an island with each area. (a) 100 square miles (b) 1500 square miles

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 \(s=10\)

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{\left(x^{2}-9\right)(2+x)}{\left(x^{2}-4\right)(3+x)}$$

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 \(r=16\)

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{\left(x^{2}-16\right)(3+x)}{\left(x^{2}-9\right)(4+x)}$$

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

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{4}-20 x^{2}+64}{x^{4}-10 x^{2}+9}$$