Chapter 7

If a ball is dropped from a high window, the distance, \(D,\) in feet, it falls is proportional to the square of the time, \(t,\) in seconds, since it was dropped and is given by $$ D=16 t^{2} $$ How far has the ball fallen after three seconds and after five seconds? Which distance is larger? Explain your answer in algebraic terms.

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \sqrt[3]{x / 8} $$

If the side length of a cube is increased by \(10 \%,\) what happens to its surface area?

A city's electricity consumption, \(E\), in gigawatt-hours per year, is given by $$ E=\frac{0.15}{p^{3 / 2}} $$ where \(p\) is the price in dollars per kilowatt-hour charged. (a) Is \(E\) a power function of \(p ?\) If so, identify the exponent and the constant of proportionality. (b) What is the electricity consumption at a price of \(\$ 0.16\) per kilowatt- hour? At a price of \(\$ 0.25\) per kilowatt hour? Explain the change in electricity consumption in algebraic terms.

In Exercises \(31-36, a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(x)=\frac{x^{3}}{3} $$

If the side length of a cube is increased by \(10 \%,\) what happens to its volume?

Without solving them, say whether the equations in Exercises \(27-42\) have (i) One positive solution (ii) One negative solution (iii) One solution at \(x=0\) (iv) Two solutions (v) \(\quad\) Three solutions (vi) No solution Give a reason for your answer. $$ x^{3}=-9 $$

The surface area of a mammal is given by \(f(M)=\) \(k M^{2 / 3},\) where \(M\) is the body mass, and the constant of proportionality \(k\) is a positive number that depends on the body shape of the mammal. Is the surface area larger for a mammal of body mass 60 kilograms or for a mammal of body mass 70 kilograms? Explain your answer in algebraic terms.

\(a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(t)=\frac{\sqrt{5}}{t^{7}} $$

The radius, \(r\), in \(\mathrm{cm}\), of a sphere of volume \(V \mathrm{~cm}^{3}\) is approximately \(r=0.620 \sqrt[3]{V}\). (a) Graph the radius function, \(r,\) for volumes from 0 to \(40 \mathrm{~cm}^{3}\). (b) Use your graph to estimate the volume of a sphere of radius \(2 \mathrm{~cm} .\)

\(a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(r)=\sqrt{\frac{12}{r}} $$

\(a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(t)=-3 t^{2}\left(-2 t^{3}\right)^{5} $$

If \(z\) is proportional to a power of \(y\) and \(y\) is proportional to a power of \(x\), is \(z\) proportional to a power of \(x\) ?

Plot the expressions \(x^{2} \cdot x^{3}, x^{5},\) and \(x^{6},\) on three separate graphs in the window \(-1

If \(z\) is proportional to a power of \(x\) and \(y\) is proportional to the same power of \(x\), is \(z+y\) proportional to a power of \(x ?\)

Without solving them, say whether the equations in Exercises \(27-42\) have (i) One positive solution (ii) One negative solution (iii) One solution at \(x=0\) (iv) Two solutions (v) \(\quad\) Three solutions (vi) No solution Give a reason for your answer. $$ x^{1 / 3}=2 $$

Plot the expressions \(-x^{4}\) and \((-x)^{4}\) on the same graph in the window \(-1

\(a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(v)=-5\left((-2 v)^{2}\right)^{3} $$

Without solving them, say whether the equations in Exercises \(27-42\) have (i) One positive solution (ii) One negative solution (iii) One solution at \(x=0\) (iv) Two solutions (v) \(\quad\) Three solutions (vi) No solution Give a reason for your answer. $$ x^{1 / 3}=-2 $$

If \(z\) is proportional to a power of \(x\) and \(y\) is proportional to a power of \(x\), is \(z y\) proportional to a power of \(x\) ?

\(a\) and \(b\) are positive numbers and \(a>b\). Which is larger, \(f(a)\) or \(f(b)\) ? $$ f(x)=2 \sqrt[3]{x}+3 \sqrt[9]{x^{3}} $$

If \(z\) is proportional to a power of \(x\) and \(y\) is proportional to a different power of \(x,\) is \(z+y\) proportional to a power of \(x ?\)

Without solving them, say whether the equations in Exercises \(27-42\) have (i) One positive solution (ii) One negative solution (iii) One solution at \(x=0\) (iv) Two solutions (v) \(\quad\) Three solutions (vi) No solution Give a reason for your answer. $$ x^{1 / 2}=12 $$

What is the exponent of the given power function? Which of (I)-(IV) in Figure 7.18 best fits its graph? Assume all constants are positive. The number of species, \(N,\) on an island as a function of the area, \(A,\) of the island: $$ N=k \sqrt[3]{A}. $$

The energy, \(E,\) in foot-pounds, delivered by an ocean wave is proportional \(^{6}\) to the length, \(L,\) of the wave times the square of its height, \(h\). (a) Write a formula for \(E\) in terms of \(L\) and \(h\). (b) A 30 -foot high wave of length 600 feet delivers 4 million foot-pounds of energy. Find the constant of proportionality and give its units. (c) If the height of a wave is one-fourth the length, find the energy \(E\) in terms of the length \(L\). (d) If the length is 5 times the height, find the energy \(E\) in terms of the height \(h\).

Poiseuille's Law tells us that the rate of flow, \(R,\) of a gas through a cylindrical pipe is proportional to the fourth power of the radius, \(r\), of the pipe, given a fixed drop in pressure between the two ends of the pipe. For a certain gas, if the rate of flow is measured in \(\mathrm{cm}^{3} / \mathrm{sec}\) and the radius is measured in \(\mathrm{cm}\), the constant of proportionality is 4.94 . (a) If the rate of flow of this gas through a pipe is 500 \(\mathrm{cm}^{3} / \mathrm{sec},\) what is the radius of the pipe? (b) Solve for the radius \(r\) in terms of the rate of flow \(R\). (c) Is \(r\) proportional to a power of \(R ?\) If so, what power?

The energy, \(E\), in foot-pounds, delivered by an ocean wave is proportional \(^{7}\) to the length, \(L,\) in feet, of the wave times the square of its height, \(h,\) in feet, with constant of proportionality 7.4 (a) If a wave is \(50 \mathrm{ft}\) long and delivers \(40,000 \mathrm{ft}\) -lbs of energy, what is its height? (b) For waves that are \(20 \mathrm{ft}\) long, solve for the height of the wave in terms of the energy. Put the answer in the form \(h=k E^{p}\) and give the values of the coefficient \(k\) and the exponent \(p\).

Without solving them, say whether the equations in Exercises \(27-42\) have (i) One positive solution (ii) One negative solution (iii) One solution at \(x=0\) (iv) Two solutions (v) \(\quad\) Three solutions (vi) No solution Give a reason for your answer. $$ x^{-2}=4 $$

A quantity \(P\) is inversely proportional to the cube of a quantity \(R\). Solve for \(R\) in terms of \(P\). Is \(R\) inversely proportional or proportional to a positive power of \(P ?\) What power?

Without solving them, say whether the equations in Exercises \(27-42\) have (i) One positive solution (ii) One negative solution (iii) One solution at \(x=0\) (iv) Two solutions (v) \(\quad\) Three solutions (vi) No solution Give a reason for your answer. $$ x^{-3}=-8 $$

What is the exponent of the given power function? Which of (I)-(IV) in Figure 7.18 best fits its graph? Assume all constants are positive. The average velocity, \(v,\) on a trip over a fixed distance \(d\) as a function of the time of travel, \(t\) $$ v=\frac{d}{t}. $$

The thrust, \(T,\) in pounds, of a ship's propeller is proportional to the square of the propeller speed, \(R\), in rotations per minute, times the fourth power of the propeller diameter, \(D,\) in feet. \(^{8}\) (a) Write a formula for \(T\) in terms of \(R\) and \(D\). (b) Solve for the propeller speed \(R\) in terms of the thrust \(T\) and the diameter \(D\). Write your answer in the form \(R=C T^{n} D^{m}\) for some constants \(C\), \(n,\) and \(m\). What are the values of \(n\) and \(m\) ? (c) Solve for the propeller diameter \(D\) in terms of the thrust \(T\) and the speed \(R\). Write your answer in the form \(D=C T^{n} R^{m}\) for some constants \(C, n,\) and \(m\). What are the values of \(n\) and \(m\) ?

Without solving them, say whether the equations in Exercises \(27-42\) have (i) One positive solution (ii) One negative solution (iii) One solution at \(x=0\) (iv) Two solutions (v) \(\quad\) Three solutions (vi) No solution Give a reason for your answer. $$ x^{-6}=-\frac{1}{64} $$

What is the exponent of the given power function? Which of (I)-(IV) in Figure 7.18 best fits its graph? Assume all constants are positive. The surface area, \(S\), of a mammal as a function of the body mass, \(B\) : $$ S=k B^{2 / 3} . $$

Without solving them, say whether the equations in Problems \(43-56\) have a positive solution \(x=a\) such that (i) \(a>1\) (ii) \(\quad a=1\) (iii) \(\quad 0

In Exercises \(43-48\), what operation transforms the first equation into the second? Identify any extraneous solutions and any solutions that are lost in the transformation. $$ \begin{aligned} \sqrt{x+4} &=x-2 \\ x+4 &=(x-2)^{2} \end{aligned} $$

What is the exponent of the given power function? Which of (I)-(IV) in Figure 7.18 best fits its graph? Assume all constants are positive. The number of animal species, \(N,\) of a certain body length as a function of the body length, \(L\) : $$ N=\frac{A}{L^{2}}. $$

What is the exponent of the given power function? Which of (I)-(IV) in Figure 7.18 best fits its graph? Assume all constants are positive. The circulation time, \(T,\) of a mammal as a function of its body mass, \(B\) : $$ T=M \sqrt[4]{B}. $$

Without solving them, say whether the equations in Problems \(43-56\) have a positive solution \(x=a\) such that (i) \(a>1\) (ii) \(\quad a=1\) (iii) \(\quad 0

In Exercises \(43-48\), what operation transforms the first equation into the second? Identify any extraneous solutions and any solutions that are lost in the transformation. $$ \begin{array}{r} t+1=1 \\ (t+1)^{2}=1 \end{array} $$

In Exercises \(43-48\), what operation transforms the first equation into the second? Identify any extraneous solutions and any solutions that are lost in the transformation. $$ \begin{aligned} r^{2}+3 r &=7 r \\ r+3 &=7 \end{aligned} $$

Without solving them, say whether the equations in Problems \(43-56\) have a positive solution \(x=a\) such that (i) \(a>1\) (ii) \(\quad a=1\) (iii) \(\quad 0

In Exercises \(43-48\), what operation transforms the first equation into the second? Identify any extraneous solutions and any solutions that are lost in the transformation. $$ \begin{aligned} (2 x)^{2} &=16 \\ 2 x &=4 \end{aligned} $$

In Exercises \(43-48\), what operation transforms the first equation into the second? Identify any extraneous solutions and any solutions that are lost in the transformation. $$ \begin{aligned} 3-\frac{1}{p} &=\frac{p-1}{p} \\ 3 p-1 &=p-1 \end{aligned} $$

In Exercises \(43-48\), what operation transforms the first equation into the second? Identify any extraneous solutions and any solutions that are lost in the transformation. $$ \begin{aligned} \frac{2 x}{x+1} &=1-\frac{2}{x+1} \\ 2 x &=x+1-2 \end{aligned} $$

The equation $$ x \sqrt{8-x}=5 $$ has two solutions. Are they positive, zero, or negative? Give an algebraic reason why this must be the case. You need not find the solutions.

Without solving them, say whether the equations in Problems \(43-56\) have a positive solution \(x=a\) such that (i) \(a>1\) (ii) \(\quad a=1\) (iii) \(\quad 0

The volume of a cone of height 2 and radius \(r\) is \(V=\frac{2}{3} \pi r^{2} .\) What is the radius of such a cone whose volume is \(3 \pi ?\)

Let \(V=s^{3}\) give the volume of a cube of side length \(s\) centimeters. For what side length is the cube's volume \(27 \mathrm{~cm}^{3} ?\)

Given that each expression is defined and not equal to zero, state its sign (positive or negative). $$ \left(1+r^{2}\right)^{2}-1 $$