Section 1
Inverse Variation
Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.$$x=1 \text { when } y=11$$
Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.$$x=-13 \text { when } y=100$$
Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.$$x=1 \text { when } y=1$$
Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.$$x=28 \text { when } y=-2$$
Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.$$x=1.2 \text { when } y=3$$
Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation.$$x=2.5 \text { when } y=100$$
Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.$$\begin{array}{|c|c|c|c|c|}\hline x & {3} & {8} & {10} & {22} \\ \hline y & {15} & {40} & {50} & {110} \\ \hline\end{array}$$
Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.$$\begin{array}{|c|c|c|c|c|}\hline x & {3} & {5} & {7} & {10.5} \\ \hline y & {14} & {8.4} & {6} & {4} \\ \hline\end{array}$$
Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.$$\begin{array}{|c|c|c|c|c|}\hline x & {0.5} & {2.1} & {3.5} & {11} \\ \hline y & {1} & {4.2} & {7} & {22} \\ \hline\end{array}$$
Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.$$\begin{array}{|c|c|c|c|c|}\hline x & {0.1} & {3} & {6} & {24} \\ \hline y & {3} & {0.1} & {0.05} & {0.0125} \\ \hline\end{array}$$
Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.$$\begin{array}{|c|c|c|c|c|}\hline x & {7} & {3} & {1} & {\frac{1}{5}} \\ \hline y & {\frac{1}{7}} & {\frac{1}{3}} & {1} & {5} \\ \hline\end{array}$$
Is the relationship between the values in each table a direct variation, an inverse variation, or neither? Write equations to model the direct and inverse variations.$$\begin{array}{|c|c|c|c|c|}\hline x & {10} & {12} & {20} & {23} \\ \hline y & {2} & {2 \frac{2}{5}} & {4} & {5 \frac{3}{5}} \\ \hline\end{array}$$
Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation and find $y$ when $x=10 .$$$x=20 \text { when } y=5$$
Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation and find $y$ when $x=10 .$$$x=20 \text { when } y=-4$$
Suppose that $x$ and $y$ vary inversely. Write a function that models each inverse variation and find $y$ when $x=10 .$$$x=5 \text { when } y=-\frac{1}{3}$$
Describe the combined variation that is modeled by each formula.$$A=\pi r^{2}$$
Describe the combined variation that is modeled by each formula.$$A=0.5 b h$$
Describe the combined variation that is modeled by each formula.$$h=\frac{2 A}{b}$$
Describe the combined variation that is modeled by each formula.$$V=\frac{B h}{3}$$
Describe the combined variation that is modeled by each formula.$$V=\pi r^{2} h$$
Describe the combined variation that is modeled by each formula.$$h=\frac{V}{\pi r^{2}}$$
Describe the combined variation that is modeled by each formula.$$V=\ell w h$$
Describe the combined variation that is modeled by each formula.$$\ell=\frac{V}{w h}$$
Write the function that models each variation. Find $z$ when $x=4$ and $y=9$$z$ varies directly with $x$ and inversely with $y .$ When $x=6$ and $y=2, z=15$
Write the function that models each variation. Find $z$ when $x=4$ and $y=9$$z$ varies jointly with $x$ and $y .$ When $x=2$ and $y=3, z=60$
Write the function that models each variation. Find $z$ when $x=4$ and $y=9$$z$ varies directly with the square of $x$ and inversely with $y .$ When $x=2$ and $y=4, z=3 .$
Write the function that models each variation. Find $z$ when $x=4$ and $y=9$$z$ varies inversely with the product of $x$ and $y .$ When $x=2$ and $y=4, z=0.5$
Write the function that models each variation. Find $z$ when $x=4$ and $y=9$a. The spreadsheet shows data that could be modeled by an equation of the form $P V=k$ . Estimate the value of $k$ .b. Estimate $P$ when $V=62$ .
Each ordered pair is from an inverse variation. Find the constant of variation.$$(6,3)$$
Each ordered pair is from an inverse variation. Find the constant of variation.$$(0.9,4)$$
Each ordered pair is from an inverse variation. Find the constant of variation.$$\left(\frac{3}{8}, \frac{2}{3}\right)$$
Each ordered pair is from an inverse variation. Find the constant of variation.$$(\sqrt{2}, \sqrt{18})$$
Each ordered pair is from an inverse variation. Find the constant of variation.$$(\sqrt{3}, \sqrt{27})$$
Each ordered pair is from an inverse variation. Find the constant of variation.$$(\sqrt{8}, \sqrt{32})$$
Mechanics Gear A drives Gear B. Gear A has $a$ teeth and speed $r_{A}$ in revolutions per minute $(r p m) .$ Gear $B$ has $b$ teeth and speed $r_{B} .$ The quantities are related by the formula $a r_{\mathrm{A}}=b r_{\mathrm{B}}$ Gear $\mathrm{A}$ has 60 teeth and speed 540 $\mathrm{rpm} .$ Gear $\mathrm{B}$ has 45 teeth. Find the speed of Gear $\mathrm{B}$ .
Physics The force $F$ of gravity on a rocket varies directly with its mass $m$ and inversely with the square of its distance $d$ from Earth. Write a model for this combined variation. $k_{d^{2}}^{m}$
Each pair of values is from a direct variation. Find the missing value.$$(3,7),(8, y)$$
Each pair of values is from a direct variation. Find the missing value.$$(2,5),(4, y)$$
Each pair of values is from a direct variation. Find the missing value.$$(4,6),(x, 3)$$
Each pair of values is from a direct variation. Find the missing value.$$(9,5),(x, 3)$$
Each pair of values is from a direct variation. Find the missing value.$$(8.3,7.1),(5, y)$$
Each pair of values is from a direct variation. Find the missing value.$$(2.6,4.5),(x, 6.3)$$
Each pair of values is from an inverse variation. Find the missing value.$$(3,7),(8, y)$$
Each pair of values is from an inverse variation. Find the missing value.$$(2,5),(4, y)$$
Each pair of values is from an inverse variation. Find the missing value.$$(4,6),(x, 3)$$
Each pair of values is from an inverse variation. Find the missing value.$$(9,5),(x, 3)$$
Each pair of values is from an inverse variation. Find the missing value.$$(8.3,7.1),(5, y)$$
Each pair of values is from an inverse variation. Find the missing value.$$(2.6,4.5),(x, 6.3)$$
Suppose that $y$ varies inversely with the square of $x,$ and $y=50$ when $x=4$ . Find $y$ when $x=5 .$
Suppose that $c$ varies jointly with $d$ and the square of $g,$ and $c=30$ when $d=15$ and $g=2 .$ Find $d$ when $c=6$ and $g=8$ .
Suppose that $d$ varies jointly with $r$ and $t,$ and $d=110$ when $r=55$ and $t=2$ . Find $r$ when $d=40$ and $t=3 .$
Construction A concrete supplier sells premixed concrete in $300-f t^{3}$ truckloads. The area $A$ that the concrete will cover is inversely proportional to the depth $d$ of the concrete.a. Write a model for the relationship between the area and the depth of a truckload of poured concrete.b. What area will the concrete cover if it is poured to a depth of 0.5 $\mathrm{ft}$ ? A depth of 1 $\mathrm{ft}$ ? A depth of 1.5 $\mathrm{ft}$ ?c. When the concrete is poured into a circular area, the depth of the concrete is inversely proportional to the square of the radius $r$ . Write a model for this relationship.
Suppose that $y$ varies directly with $x$ and inversely with $z^{2},$ and $x=48$ when $y=8$ and $z=3 .$ Find $x$ when $y=12$ and $z=2$ .
Suppose that $t$ varies directly with $s$ and inversely with the square of $r .$ How is the value of $t$ changed when the value of $s$ is doubled? Is tripled?
Suppose that $x$ varies directly with the square of $y$ and inversely with $z .$ How is the value of $x$ changed if the value of $y$ is halved? Is quartered?
Writing. Explain why 0 cannot be in the domain of an inverse variation.
Critical Thinking. Suppose that $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ are values from an inverse variation. Show that $\frac{x_{1}}{x_{2}}=\frac{y_{2}}{y_{1}} .$
Open-Ended. The height $h$ of a cylinder varies directly with its volume $V$ and inversely with the square of its radius $r .$ Find at least four ways to change the volume and radius of a cylinder so that its height is quadrupled.
Health Health care professionals use the body mass index (BMI) to establish guidelines for determining any possible risk of their patients and for planing any useful preventative programs. The BMI varies directly with weight and inversely with the square of height. Use this portion of the BMI chart to determine the BMI formula.
Which equation does NOT represent inverse variation between $x$ and $z ?$$$\begin{array}{ll}{\text { A. } x=\frac{y}{z}} & {\text { B. } x=\frac{-15 z}{y}} \\ {\text { C. } z=\frac{-15 y}{x}} & {\text { D. } x z=5 y}\end{array}$$
If $p$ and $q$ vary inversely, and $p=10$ when $q=-4,$ what is $q$ when $p=-2 ?$$$\begin{array}{lllll}{\text { F. } 20} & {\text { G. } \frac{4}{5}} & {\text { H. }-\frac{4}{5}} & {\text { 1. }-20}\end{array}$$
Which equation shows that $z$ varies directly with the square of $x$ and inversely with the cube of $y ?$$$\begin{array}{llll}{\text { A. } z=\frac{x^{2}}{y^{3}}} & {\text { B. } z=\frac{x^{3}}{y^{2}}} & {\text { C. } z=\frac{y^{2}}{x^{3}}} & {\text { D. } z=\frac{y^{3}}{x^{2}}}\end{array}$$
Describe how the variables $A$ and $r$ vary in the formula for the area of a circle, $A=\pi r^{2} .$
Which data set shows inverse variation: $(24.4,4.8)$ and $(9.6,12.2),$ or $(24.0,4.5)$ and $(18.0,6.5) ?$ Explain.
Solve each equation.$$\ln 4+\ln x=5$$
Solve each equation.$$\ln x-\ln 3=4$$
Solve each equation.$$2 \ln x+3 \ln 4=4$$
Multiply and simplify. Assume that all variables are positive.$$-5 \sqrt{6 x} \cdot 3 \sqrt{6 x^{2}}$$
Multiply and simplify. Assume that all variables are positive.$$3 \sqrt[3]{4 x^{2}} \cdot 7 \sqrt[3]{12 x^{4}}$$
Multiply and simplify. Assume that all variables are positive.$$\sqrt{5 x^{3}} \cdot \sqrt{40 x y^{7}}$$
Simplify each radical expression. Use absolute value bars where they are needed.$$\sqrt{x^{10} y^{100}}$$
Simplify each radical expression. Use absolute value bars where they are needed.$$\sqrt[3]{-64 a^{3} b^{6}}$$
Simplify each radical expression. Use absolute value bars where they are needed.$$\sqrt[4]{64 m^{8} n^{4}}$$
Simplify each radical expression. Use absolute value bars where they are needed.$$\sqrt[4]{x^{4}}$$