Chapter 4

Market Research A market analyst working for a small- appliance manufacturer finds that if the firm produces and sells \(x\) blenders annually, the total profit (in dollars) is $$P(x)=8 x+0.3 x^{2}-0.0013 x^{3}-372$$ Graph the function \(P\) in an appropriate viewing rectangle and use the graph to answer the following questions. (a) When just a few blenders are manufactured, the firm loses money (profit is negative). (For example, \(P(10)=-263.3\) so the firm loses \(\$ 263.30\) if it produces and sells only 10 blenders.) How many blenders must the firm produce to break even? (b) Does profit increase indefinitely as more blenders are produced and sold? If not, what is the largest possible profit the firm could have?

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=6 x^{4}-7 x^{3}-8 x^{2}+5 x $$

Graph the rational function, and find all vertical asymptotes, x- and y-intercepts, and local extrema, correct to the nearest decimal. Then use long division to find a polynomial that has the same end behavior as the rational function, and graph both functions in a sufficiently large viewing rectangle to verify that the end behaviors of the polynomial and the rational function are the same. $$ r(x)=\frac{4+x^{2}-x^{4}}{x^{2}-1} $$

Population Change The rabbit population on a small island is observed to be given by the function $$P(t)=120 t-0.4 t^{4}+1000$$ where \(t\) is the time (in months) since observations of the island began. (a) When is the maximum population attained, and what is that maximum population? (b) When does the rabbit population disappear from the island? (GRAPH NOT COPY)

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24 $$

Population Growth Suppose that the rabbit population on Mr. Jenkins' farm follows the formula $$ p(t)=\frac{3000 t}{t+1} $$ where \(t \geq 0\) is the time (in months) since the beginning of the year. (a) Draw a graph of the rabbit population. (b) What eventually happens to the rabbit population?

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=8 x^{5}-14 x^{4}-22 x^{3}+57 x^{2}-35 x+6 $$

Drug Concentration After a certain drug is injected into a patient, the concentration \(c\) of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in minutes since the injection), the concentration (in mg/L) is given by $$ c(t)=\frac{30 t}{t^{2}+2} $$ (a) Draw a graph of the drug concentration. (b) What eventually happens to the concentration of drug in the bloodstream?

Show that the polynomial does not have any rational zeros. $$ P(x)=x^{3}-x-2 $$

Drug Concentration \(\mathrm{A}\) drug is administered to a patient, and the concentration of the drug in the bloodstream is monitored. At time \(t \geq 0\) (in hours since giving the drug), the concentration (in \(\mathrm{mg} / \mathrm{L}\) ) is given by $$ c(t)=\frac{5 t}{t^{2}+1} $$ Graph the function \(c\) with a graphing device. (a) What is the highest concentration of drug that is reached in the patient's bloodstream? (b) What happens to the drug concentration after a long period of time? (c) How long does it take for the concentration to drop below 0.3 \(\mathrm{mg} / \mathrm{L} ?\)

Graphs of Large Powers Graph the functions \(y=x^{2}\) , \(y=x^{3}, y=x^{4}\) , and \(y=x^{5},\) for \(-1 \leq x \leq 1\) , on the same coordinate axes. What do you think the graph of \(y=x^{100}\) ? would look like on this same interval? What about \(y=x^{101} ?\) Make a table of values to confirm your answers.

Show that the polynomial does not have any rational zeros. $$ P(x)=2 x^{4}-x^{3}+x+2 $$

Flight of a Rocket Suppose a rocket is fired upward from the surface of the earth with an initial velocity \(v\) (measured in meters per second). Then the maximum height \(h\) (in meters) reached by the rocket is given by the function $$ H(v)=\frac{R v^{2}}{2 g R-v^{2}} $$ where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of the earth and \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity. Use a graphing device to draw a graph of the function \(h .\) (Note that \(h\) and \(v\) must both be positive, so the viewing rectangle need not contain negative values.) What does the vertical asymptote represent physically?

Show that the polynomial does not have any rational zeros. $$ P(x)=3 x^{3}-x^{2}-6 x+12 $$

The Doppler Effect As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the ob- server than it would if the train were at rest, because the crests of the sound waves are compressed closer together. This phenomenon is called the Doppler effect. The observed pitch \(P\) is a function of the speed \(v\) of the train and is given by $$ P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right) $$ where \(P_{0}\) is the actual pitch of the whistle at the source and \(S_{0}=332 \mathrm{m} / \mathrm{s}\) is the speed of sound in air. Suppose that a train has a whistle pitched at \(P_{0}=440 \mathrm{Hz}\) . Graph the func- tion \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically?

Show that the polynomial does not have any rational zeros. $$ P(x)=x^{50}-5 x^{25}+x^{2}-1 $$

Focusing Distance For a camera with a lens of fixed fo- cal length \(F\) to focus on an object located a distance \(x\) from the lens, the film must be placed a distance \(y\) behind the lens, where \(F, x,\) and \(y\) are related by $$ \frac{1}{x}+\frac{1}{y}=\frac{1}{F} $$ (See the figure.) Suppose the camera has a \(55-\mathrm{mm}\) lens \((F=55) .\) (a) Express \(y\) as a function of \(x\) and graph the function. (b) What happens to the focusing distance \(y\) as the object moves far away from the lens? (c) What happens to the focusing distance \(y\) as the object moves close to the lens?

Impossible Situation? Is it possible for a polynomial to have two local maxima and no local minimum? Explain.

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ x^{3}-3 x^{2}-4 x+12=0 ; \quad[-4,4] \text { by }[-15,15] $$

Constructing a Rational Function from Its Asymptotes Give an example of a rational function that has vertical asymptote \(x=3\) . Now give an example of one that has vertical asymptote \(x=3\) and horizontal asymptote \(y=2\) Now give an example of a rational function with vertical asymptotes \(x=1\) and \(x=-1,\) horizontal asymptote \(y=0,\) and \(x\) -intercept \(4 .\)

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ x^{4}-5 x^{2}+4=0 ;[-4,4] \mathrm{by}[-30,30] $$

A Rational Function with No Asymptote Explain how you can tell (without graphing it) that the function $$ r(x)=\frac{x^{6}+10}{x^{4}+8 x^{2}+15} $$ has no \(x\) -intercept and no horizontal, vertical, or slant asymptote. What is its end behavior?'

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ 2 x^{4}-5 x^{3}-14 x^{2}+5 x+12=0 ; \quad[-2,5] \text { by }[-40,40] $$

Graphs with Holes In this chapter we adopted the convention that in rational functions, the numerator and denominator don't share a common factor. In this exercise we consider the graph of a rational function that does not satisfy this rule. (a) Show that the graph of $$ r(x)=\frac{3 x^{2}-3 x-6}{x-2} $$ is the line \(y=3 x+3\) with the point \((2,9)\) removed. [Hint. Factor. What is the domain of \(r ? ]\) (b) Graph the rational functions: $$ \begin{aligned} s(x) &=\frac{x^{2}+x-20}{x+5} \\ t(x) &=\frac{2 x^{2}-x-1}{x-1} \\ u(x) &=\frac{x-2}{x^{2}-2 x} \end{aligned} $$

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ 3 x^{3}+8 x^{2}+5 x+2=0 ;[-3,3] \text { by }[-10,10] $$

Transformations of \(y=1 / x^{2}\) In Example 2 we saw that some simple rational functions can be graphed by shifting, stretching, or reflecting the graph of \(y=1 / x\) . In this exercise we consider rational functions that can be graphed by transforming the graph of \(y=1 / x^{2},\) shown on the following page. (a) Graph the function $$ r(x)=\frac{1}{(x-2)^{2}} $$ by transforming the graph of \(y=1 / x^{2}\) . (b) Use long division and factoring to show that the function $$ s(x)=\frac{2 x^{2}+4 x+5}{x^{2}+2 x+1} $$ can be written as $$ s(x)=2+\frac{3}{(x+1)^{2}} $$ Then graph \(s\) by transforming the graph of \(y=1 / x^{2} .\) (c) One of the following functions can be graphed by trans- forming the graph of \(y=1 / x^{2} ;\) the other cannot. Use transformations to graph the one that can be, and explain why this method doesn't work for the other one. $$ p(x)=\frac{2-3 x^{2}}{x^{2}-4 x+4} \quad q(x)=\frac{12 x-3 x^{2}}{x^{2}-4 x+4} $$

Use a graphing device to find all real solutions of the equation, rounded to two decimal places. $$ x^{4}-x-4=0 $$

Use a graphing device to find all real solutions of the equation, rounded to two decimal places. $$ 2 x^{3}-8 x^{2}+9 x-9=0 $$

Use a graphing device to find all real solutions of the equation, rounded to two decimal places. $$ 4.00 x^{4}+4.00 x^{3}-10.96 x^{2}-5.88 x+9.09=0 $$

Use a graphing device to find all real solutions of the equation, rounded to two decimal places. $$ x^{5}+2.00 x^{4}+0.96 x^{3}+5.00 x^{2}+10.00 x+4.80=0 $$

Show that the equation $$ x^{5}-x^{4}-x^{3}-5 x^{2}-12 x-6=0 $$ has exactly one rational root, and then prove that it must have either two or four irrational roots.

Volume of a Silo \(A\) grain silo consists of a cylindrical main section and a hemispherical roof. If the total volume of the silo (including the part inside the roof section) is \(15,000 \mathrm{ft}^{3}\) and the cylindrical part is 30 \(\mathrm{ft}\) tall, what is the radius of the silo, rounded to the nearest tenth of a foot?

Dimensions of a Lot A rectangular parcel of land has an area of 5000 \(\mathrm{ft}^{2}\) . A diagonal between opposite comers is measured to be 10 \(\mathrm{ft}\) longer than one side of the parcel. What are the dimensions of the land, rounded to the nearest foot?

Volume of a Box An open box with a volume of 1500 \(\mathrm{cm}^{3}\) is to be constructed by taking a piece of cardboard 20 \(\mathrm{cm}\) by \(40 \mathrm{cm},\) cutting squares of side length \(x \mathrm{cm}\) from each comer, and folding up the sides. Show that this can be done in two different ways, and find the exact dimensions of the box in each case.

Volume of a Rocket A rocket consists of a right circular cylinder of height 20 \(\mathrm{m}\) surmounted by a cone whose height and diameter are equal and whose radius is the same as that of the cylindrical section. What should this radius be (rounded to two decimal places) if the total volume is to be 500\(\pi / 3 \mathrm{m}^{3}\) ?

Volume of a Box A rectangular box with a volume of 2\(\sqrt{2} \mathrm{ft}^{3}\) has a square base as shown below. The diagonal of the box (between a pair of opposite comers) is 1 ft longer than each side of the base. (a) If the base has sides of length \(x\) feet, show that $$ x^{4}-2 x^{5}-x^{4}+8=0 $$ (b) Show that two different boxes satisfy the given conditions. Find the dimensions in each case, rounded to the nearest hundredth of a foot.

A box with a square base has length plus girth of 108 in. (Girth is the distance "around" the box.) What is the length of the box if its volume is 2200 in \(^{3} ?\)

How Many Real Zeros Can a Polynomial Have? Give examples of polynomials that have the following properties, or explain why it is impossible to find such a polynomial. (a) A polynomial of degree 3 that has no real zeros (b) A polynomial of degree 4 that has no real zeros (c) A polynomial of degree 3 that has three real zeros, only one of which is rational (d) \(\mathrm{A}\) polynomial of degree 4 that has four real zeros, none of which is rational What must be true about the degree of a polynomial with integer coefficients if it has no real zeros?