Chapter 7

Write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ \sqrt{\frac{\sqrt{3}}{4} s} $$

In Exercises \(13-16,\) what happens to \(y\) when \(x\) is doubled? Here \(k\) is a positive constant. $$ \frac{y}{x^{4}}=k $$

In Exercises \(1-21,\) solve the equation for the variable. $$ \frac{1}{4} t^{3}=\frac{4}{t} $$

Write the expression as a constant times a power of a variable. Identify the coefficient and the exponent. $$ \sqrt{\sqrt{4 t^{3}}} $$

(a) Is \(y\) proportional, or is it inversely proportional, to a positive power of \(x\) ? (b) Make a table of values showing corresponding values for \(y\) when \(x\) is \(1,10,100,\) and 1000 . (c) Use your table to determine whether \(y\) increases or decreases as \(x\) gets larger. \(y=\frac{1}{x}\)

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. \({ }^{2}\) (a) Write a formula for \(T\) in terms of \(R\) and \(D\). (b) If \(R=300 D\) for a certain propeller, is \(T\) a power function of \(D ?\) (c) If \(D=0.25 \sqrt{R}\) for a different propeller, is \(T\) a power function of \(R\) ?

In Exercises \(1-21,\) solve the equation for the variable. $$ 16-\frac{1}{L^{2}}=0 $$

In Exercises \(17-30\), can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \frac{2}{3 \sqrt{x}} $$

(a) Is \(y\) proportional, or is it inversely proportional, to a positive power of \(x\) ? (b) Make a table of values showing corresponding values for \(y\) when \(x\) is \(1,10,100,\) and 1000 . (c) Use your table to determine whether \(y\) increases or decreases as \(x\) gets larger. \(y=\frac{5}{x^{2}}\)

Poiseuille's Law gives the rate of flow, \(R,\) of a gas through a cylindrical pipe in terms of the radius of the pipe, \(r,\) for a fixed drop in pressure between the two ends of the pipe. (a) Find a formula for Poiseuille's Law, given that the rate of flow is proportional to the fourth power of the radius. (b) If \(R=400 \mathrm{~cm}^{3} / \mathrm{sec}\) in a pipe of radius \(3 \mathrm{~cm}\) for a certain gas, find a formula for the rate of flow of that gas through a pipe of radius \(r \mathrm{~cm}\). (c) What is the rate of flow of the gas in part (b) through a pipe with a \(5 \mathrm{~cm}\) radius?

In Exercises \(1-21,\) solve the equation for the variable. $$ \sqrt{r^{2}+144}=13 $$

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \frac{1}{3 x^{2}} $$

The circulation time of a mammal (that is, the average time it takes for all the blood in the body to circulate once and return to the heart) is proportional to the fourth root of the body mass of the mammal. (a) Write a formula for the circulation time, \(T,\) in terms of the body mass, \(B\). (b) If an elephant of body mass 5230 kilograms has a circulation time of 148 seconds, find the constant of proportionality. (c) What is the circulation time of a human with body mass 70 kilograms?

In Exercises \(1-21,\) solve the equation for the variable. $$ \frac{1}{\sqrt[3]{x}}=-3 $$

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \left(7 x^{3}\right)^{2} $$

When an aircraft takes off, it accelerates until it reaches its takeoff speed \(V\). In doing so it uses up a distance \(R\) of the runway, where \(R\) is proportional to the square of the takeoff speed. If \(V\) is measured in mph and \(R\) is measured in feet, then 0.1639 is the constant of proportionality. (a) A Boeing \(747-400\) aircraft has a takeoff speed of about 210 miles per hour. How much runway does it need? (b) What would the constant of proportionality be if \(R\) was measured in meters, and \(V\) was measured in meters per second?

In Exercises \(1-21,\) solve the equation for the variable. $$ 4 \sqrt{x}-2 \sqrt{x}=\frac{2}{3} x $$

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \sqrt{\frac{49}{x^{5}}} $$

Biologists estimate that the number of animal species of a certain body length is inversely proportional to the square of the body length. \(^{4}\) Write a formula for the number of animal species, \(N,\) of a certain body length in terms of the length, \(L\). Are there more species at large lengths or at small lengths? Explain.

In Exercises \(1-21,\) solve the equation for the variable. $$ 12=\sqrt{\frac{z}{5 \pi}} $$

In Problems \(21-24, a\) and \(b\) are positive constants. If \(a>b\) then which is larger? \(a^{4}, b^{4}\)

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \left(3 x^{2}\right)^{3} $$

In Problems \(22-25,\) find possible formulas for the power functions. $$ \begin{array}{l|l|l|l|c} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 0 & 2 & 8 & 18 \\ \hline \end{array} $$

In Exercises 22-26, solve the equation for the indicated variable. Assume all other letters represent nonzero constants. $$ y=k x^{2}, \text { for } x $$

\(a\) and \(b\) are positive constants. If \(a>b\) then which is larger? \(a^{1 / 4}, b^{1 / 4}\)

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \left(\frac{2}{\sqrt{x}}\right)^{3} $$

In Problems \(22-25,\) find possible formulas for the power functions. $$ \begin{array}{l|l|c|c|c} \hline x & 1 & 2 & 3 & 4 \\ \hline y & 4 & 16 & 36 & 64 \\ \hline \end{array} $$

In Exercises 22-26, solve the equation for the indicated variable. Assume all other letters represent nonzero constants. $$ A=\frac{1}{2} \pi r^{2}, \text { for } r $$

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ y=\frac{1}{5 x} $$

In Problems \(22-25,\) find possible formulas for the power functions. $$ \begin{array}{c|c|c|c|c} \hline x & -2 & -1 & 1 & 2 \\ \hline y & -16 & -1 & -1 & -16 \\ \hline \end{array} $$

In Exercises 22-26, solve the equation for the indicated variable. Assume all other letters represent nonzero constants. $$ L=k B^{2} D^{3}, \text { for } D $$

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \left(x^{2}+3 x^{2}\right)^{2} $$

(a) In Figure 7.7 , for which \(x\) -values is the graph of \(y=x^{5}\) above the graph of \(y=x^{3},\) and for which \(x\) -values is it below? (b) Express your answers in part (a) algebraically using inequalities.

In Problems \(22-25,\) find possible formulas for the power functions. $$ \begin{array}{c|c|c|c|c} \hline x & -2 & -1 & 1 & 2 \\ \hline y & 8 / 5 & 1 / 5 & -1 / 5 & -8 / 5 \\ \hline \end{array} $$

In Exercises 22-26, solve the equation for the indicated variable. Assume all other letters represent nonzero constants. $$ y^{2} x^{2}=\left(3 y^{2}\right)^{2}, \text { for } x $$

The volume \(V\) of a sphere is a function of its radius \(r\) given by $$ V=f(r)=\frac{4}{3} \pi r^{3} $$ (a) Find \(\frac{f(2 r)}{f(r)}\). $$ \text { (b) Find } \frac{f(r)}{f\left(\frac{1}{2} r\right)} \text { . } $$ (c) What do you notice about your answers to (a) and (b)? Explain this result in terms of sphere volumes.

In Exercises 22-26, solve the equation for the indicated variable. Assume all other letters represent nonzero constants. $$ w=4 \pi \sqrt{\frac{x}{t}}, \text { for } x $$

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \sqrt{9 x^{5}} $$

The gravitational force \(F\) exerted on an object of mass \(m\) at a distance \(r\) from the Earth's center is given by $$ \begin{array}{l} \quad F=g(r)=\mathrm{kmr}^{-2}, \quad k, m \text { constant. } \\ \text { (a) Find } \frac{g\left(\frac{r}{10}\right)}{g(r)} . \quad \text { (b) Find } \frac{g(r)}{g(10 r)} . \end{array} $$ (c) What do you notice about your answers to (a) and (b)? Explain this result in terms of gravitational force.

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. $$ x^{3}=5 $$

Figure 7.6 shows that the graph of \(y=x^{4}\) is above the graph of \(y=x^{2}\) when \(x\) is greater than \(1,\) and it is below it when \(x\) is between 0 and \(1 .\) Express these facts algebraically using inequalities.

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \left(\frac{1}{2 \sqrt{x}}\right)^{3} $$

A square of side \(x\) has area \(x^{2} .\) By what factor does the area change if the length is (a) Doubled? (b) Tripled? (c) Halved? (d) Multiplied by \(0.1 ?\)

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. $$ x^{5}=3 $$

A cube of side \(x\) has volume \(x^{3} .\) By what factor does the volume change if the length is (a) Doubled? (b) Tripled? (c) Halved? (d) \(\quad\) Multiplied by \(0.1 ?\)

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. $$ x^{2}=5 $$

A student takes a part-time job to earn $$\$ 2400$$ for summer travel. The number of hours, \(h,\) the student has to work is inversely proportional to the wage, \(w\), in dollars per hour, and is given by $$ h=\frac{2400}{w} $$ (a) How many hours does the student have to work if the job pays $$\$ 4$$ an hour? What if it pays $$\$ 10$$ an hour? (b) How do the number of hours change as the wage goes up from $$\$ 4$$ an hour to $$\$ 10$$ an hour? Explain your answer in algebraic and practical terms. (c) Is the wage, \(w\), inversely proportional to the number hours, \(h\) ? Express \(w\) as a function of \(h\).

Can the expression be written in the form \(k x^{p}\) ? If so, give the values of \(k\) and \(p\). $$ \left((-2 x)^{2}\right)^{3} $$

If the radius of a circle is halved, what happens to its area?

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. $$ x^{2}=0 $$