Chapter 4

The period \(P\) of a simple pendulum that is, the time required for one complete oscillation-is directly proportional to the square root of its length \(l\). (a) Express \(P\) in terms of \(l\) and a constant of proportionality \(k\). (b) If a pendulum 2 feet long has a period of \(1.5\) seconds, find the value of \(k\) in part (a). (c) Find the period of a pendulum 6 feet long.

Sketch the graph of \(f\). $$ f(x)=\frac{-4}{(x-2)^{2}} $$

Exer. \(15-24\) : Find all solutions of the equation. $$ 2 x^{3}-3 x^{2}-17 x+30=0 $$

Find a polynomial \(f(x)\) with leading coefficient 1 and having the given degree and zeros. degree \(3 ; \quad\) zeros \(-2,0,5\)

A circular cylinder is sometimes used in physiology as a simple representation of a human limb. (a) Express the volume \(V\) of a cylinder in terms of its length \(L\) and the square of its circumference \(C\). (b) The formula obtained in part (a) can be used to approximate the volume of a limb from length and circumference measurements. Suppose the (average) circumference of a human forearm is 22 centimeters and the average length is 27 centimeters. Approximate the volume of the forearm to the nearest \(\mathrm{cm}^{3}\).

Sketch the graph of \(f\). $$ f(x)=\frac{2}{(x+1)^{2}} $$

Find a polynomial \(f(x)\) with leading coefficient 1 and having the given degree and zeros. degree \(3 ; \quad\) zeros \(\pm 2,3\)

Kepler's third law states that the period \(T\) of a planet (the time needed to make one complete revolution about the sun) is directly proportional to the \(\frac{3}{2}\) power of its average distance \(d\) from the sun. (a) Express \(T\) as a function of \(d\) by means of a formula that involves a constant of proportionality \(k\). (b) For the planet Earth, \(T=365\) days and \(d=93\) million miles. Find the value of \(k\) in part (a). (c) Estimate the period of Venus if its average distance from the sun is 67 million miles.

Sketch the graph of \(f\). $$ f(x)=\frac{x-3}{x^{2}-1} $$

Exer. \(15-24\) : Find all solutions of the equation. $$ x^{4}+3 x^{3}-30 x^{2}-6 x+56=0 $$

Find a polynomial \(f(x)\) with leading coefficient 1 and having the given degree and zeros. degree \(4 ; \quad\) zeros \(-2, \pm 1,4\)

It is known from physics that the range \(R\) of a projectile is directly proportional to the square of its velocity \(v\). (a) Express \(R\) as a function of \(v\) by means of a formula that involves a constant of proportionality \(k\). (b) A motorcycle daredevil has made a jump of 150 feet. If the speed coming off the ramp was \(70 \mathrm{mi} / \mathrm{hr}\), find the value of \(k\) in part (a). (c) If the daredevil can reach a speed of \(80 \mathrm{mi} / \mathrm{hr}\) coming off the ramp and maintain proper balance, estimate the possible length of the jump.

Sketch the graph of \(f\). $$ f(x)=\frac{x+4}{x^{2}-4} $$

Exer. \(15-24\) : Find all solutions of the equation. $$ 3 x^{5}-10 x^{4}-6 x^{3}+24 x^{2}+11 x-6=0 $$

Find a polynomial \(f(x)\) with leading coefficient 1 and having the given degree and zeros. degree \(4 ; \quad\) zeros \(-3,0,1,5\)

The speed \(V\) at which an automobile was traveling before the brakes were applied can sometimes be estimated from the length \(L\) of the skid marks. Assume that \(V\) is directly proportional to the square root of \(L\). (a) Express \(V\) as a function of \(L\) by means of a formula that involves a constant of proportionality \(k\). (b) For a certain automobile on a dry surface, \(L=50 \mathrm{ft}\) when \(V=35 \mathrm{mi} / \mathrm{hr}\). Find the value of \(k\) in part (a). (c) Estimate the initial speed of the automobile in part (b) if the skid marks are 150 feet long.

Sketch the graph of \(f\). $$ f(x)=\frac{2 x^{2}-2 x}{x^{2}+x} $$

Exer. \(15-24\) : Find all solutions of the equation. $$ 6 x^{5}+19 x^{4}+x^{3}-6 x^{2}=0 $$

Coulomb's law in electrical theory states that the force \(F\) of attraction between two oppositely charged particles varies directly as the product of the magnitudes \(Q_{1}\) and \(Q_{2}\) of the charges and inversely as the square of the distance \(d\) between the particles. (a) Find a formula for \(F\) in terms of \(Q_{1}, Q_{2}, d\), and a constant of variation \(k\). (b) What is the effect of reducing the distance between the particles by a factor of one-fourth?

Sketch the graph of \(f\). $$ f(x)=\frac{-3 x^{2}-3 x+6}{x^{2}-9} $$

Exer. \(15-24\) : Find all solutions of the equation. $$ 6 x^{4}+5 x^{3}-17 x^{2}-6 x=0 $$

Threshold weight \(W\) is defined to be that weight beyond which risk of death increases significantly. For middle-aged males, \(W\) is directly proportional to the third power of the height \(h\). (a) Express \(W\) as a function of \(h\) by means of a formula that involves a constant of proportionality \(k\). (b) For a 6-foot male, \(W\) is about 200 pounds. Find the value of \(k\) in part (a). (c) Estimate, to the nearest pound, the threshold weight for an individual who is 5 feet 6 inches tall.

Sketch the graph of \(f\). $$ f(x)=\frac{-x^{2}-x+6}{x^{2}+3 x-4} $$

Exer. \(15-24\) : Find all solutions of the equation. $$ 8 x^{3}+18 x^{2}+45 x+27=0 $$

The ideal gas law states that the volume \(V\) that a gas occupies is directly proportional to the product of the number \(n\) of moles of gas and the temperature \(T\) (in \(\mathrm{K}\) ) and is inversely proportional to the pressure \(P\) (in atmospheres). (a) Express \(V\) in terms of \(n, T, P\), and a constant of proportionality \(k\). (b) What is the effect on the volume if the number of moles is doubled and both the temperature and the pressure are reduced by a factor of one-half?

Sketch the graph of \(f\). $$ f(x)=\frac{x^{2}-3 x-4}{x^{2}+x-6} $$

Exer. \(15-24\) : Find all solutions of the equation. $$ 3 x^{3}-x^{2}+11 x-20=0 $$

Poiseuille's law states that the blood flow rate \(F\) (in \(L / \mathrm{min}\) ) through a major artery is directly proportional to the product of the fourth power of the radius \(r\) of the artery and the blood pressure \(P\). (a) Express \(F\) in terms of \(P, r\), and a constant of proportionality \(k\). (b) During heavy exercise, normal blood flow rates sometimes triple. If the radius of a major artery increases by \(10 \%\), approximately how much harder must the heart pump?

Sketch the graph of \(f\). $$ f(x)=\frac{3 x^{2}-3 x-36}{x^{2}+x-2} $$

Suppose 200 trout are caught, tagged, and released in a lake's general population. Let \(T\) denote the number of tagged fish that are recaptured when a sample of \(n\) trout are caught at a later date. The validity of the markrecapture method for estimating the lake's total trout population is based on the assumption that \(T\) is directly proportional to \(n\). If 10 tagged trout are recovered from a sample of 300 , estimate the total trout population of the lake.

Sketch the graph of \(f\). $$ f(x)=\frac{2 x^{2}+4 x-48}{x^{2}+3 x-10} $$

When uranium disintegrates into lead, one step in the process is the radioactive decay of radium into radon gas. Radon enters through the soil into home basements, where it presents a health hazard if inhaled. In the simplest case of radon detection, a sample of air with volume \(V\) is taken. After equilibrium has been established, the radioactive decay \(D\) of the radon gas is counted with efficiency \(E\) over time \(t\). The radon concentration \(C\) present in the sample of air varies directly as the product of \(D\) and \(E\) and inversely as the product of \(V\) and \(t\).

Sketch the graph of \(f\). $$ f(x)=\frac{-2 x^{2}+10 x-12}{x^{2}+x} $$

Does there exist a polynomial of degree 3 with real coefficients that has zeros \(1,-1\), and \(i\) ? Justify your answer.

Sketch the graph of \(f\). $$ f(x)=\frac{2 x^{2}+8 x+6}{x^{2}-2 x} $$

The polynomial \(f(x)=x^{3}-i x^{2}+2 i x+2\) has the complex number \(i\) as a zero; however, the conjugate \(-i\) of \(i\) is not a zero. Why doesn't this result contradict the theorem on conjugate pair zeros of a polynomial?

A thin flat plate is situated in an \(x y\)-plane such that the density \(d\) (in \(\mathrm{lb} / \mathrm{ft}^{2}\) ) at the point \(P(x, y)\) is inversely proportional to the square of the distance from the origin. What is the effect on the density at \(P\) if the \(x\) - and \(y\)-coordinates are each multiplied by \(\frac{1}{3} ?\)

If \(n\) is an odd positive integer, prove that a polynomial of degree \(n\) with real coefficients has at least one real zero.

A flat metal plate is positioned in an \(x y\)-plane such that the temperature \(T\left(\right.\) in \({ }^{\circ} \mathrm{C}\) ) at the point \((x, y)\) is inversely proportional to the distance from the origin. If the temperature at the point \(P(3,4)\) is \(20^{\circ} \mathrm{C}\), find the temperature at the point \(Q(24,7)\).

Sketch the graph of \(f\). $$ f(x)=\frac{x^{2}-2 x+1}{x^{3}-9 x} $$

If a polynomial of the form $$ x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}, $$ where each \(a_{k}\) is an integer, has a rational root \(r\), show that \(r\) is an integer and is a factor of \(a_{0}\).

Examine the expression for the given set of data points of the form \((x, y)\). Find the constant of variation and a formula that describes how \(y\) varies with respect to \(x\). $$ \begin{aligned} y / x ; &\\{(0.6,0.72),(1.2,1.44),(4.2,5.04),(7.1,8.52)\\\ &(9.3,11.16)\\} \end{aligned} $$

Sketch the graph of \(f\). $$ f(x)=\frac{-3 x^{2}}{x^{2}+1} $$

Examine the expression for the given set of data points of the form \((x, y)\). Find the constant of variation and a formula that describes how \(y\) varies with respect to \(x\). $$ \begin{aligned} &\\{(0.2,-26.5),(0.4,-13.25),(0.8,-6.625) \\ &(1.6,-3.3125),(3.2,-1.65625)\\} \end{aligned} $$

Sketch the graph of \(f\). $$ f(x)=\frac{x^{2}-4}{x^{2}+1} $$

The frame for a shipping crate is to be constructed from 24 feet of \(2 \times 2\) lumber. Assuming the crate is to have square ends of length \(x\) feet, determine the value(s) of \(x\) that result(s) in a volume of \(4 \mathrm{ft}^{3}\). (See Exercise 42 of Section 4.1.)

Examine the expression for the given set of data points of the form \((x, y)\). Find the constant of variation and a formula that describes how \(y\) varies with respect to \(x\). $$ \begin{aligned} &\\{(0.16,-394.53125),(0.8,-15.78125) \\ &(1.6,-3.9453125),(3.2,-0.986328125)\\} \end{aligned} $$

A right triangle has area \(30 \mathrm{ft}^{2}\) and a hypotenuse that is 1 foot longer than one of its sides. (a) If \(x\) denotes the length of this side, then show that \(2 x^{3}+x^{2}-3600=0 .\) (b) Show that there is a positive root of the equation in part (a) and that this root is less than \(13 .\) (c) Find the lengths of the sides of the triangle.

Examine the expression for the given set of data points of the form \((x, y)\). Find the constant of variation and a formula that describes how \(y\) varies with respect to \(x\). $$ \begin{aligned} y / x^{3} ; &\\{(0.11,0.00355377),(0.56,0.46889472)\\\ &(1.2,4.61376),(2.4,36.91008)\\} \end{aligned} $$

Find the oblique asymptote, and sketch the graph of \(f\). $$ f(x)=\frac{2 x^{2}-x-3}{x-2} $$