Answer Key 2.7

$x=ky$
$x=kyz$
$x=\dfrac{k}{y}$
$x=ky^2$
$x=kzy$
$x=\dfrac{k}{y^3}$
$x=ky^2\sqrt{z}$
$x=\dfrac{k}{y^6}$
$x=\dfrac{ky^3}{\sqrt{z}}$
$x=\dfrac{k}{y^2\sqrt{z}}$
$x=\dfrac{kzy}{p^3}$
$x=\dfrac{k}{y^3z^2}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ A&=&kB \\ (15)&=&k(5) \\ \\ \dfrac{15}{5}&=&\dfrac{k(5)}{5} \\ \\ k&=&3 \end{array}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ P&=&kQR \\ (12)&=&k(8)(3) \\ \\ \dfrac{12}{24}&=&\dfrac{k(8)(3)}{24} \\ \\ k&=&\dfrac{1}{2} \end{array}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ \\ \\ A&=&\dfrac{k}{B} \\ \\ (7)&=&\dfrac{k}{(4)} \\ \\ (4)7&=&\dfrac{k}{\cancel{4}}\cancel{(4)} \\ \\ k&=&28 \end{array}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ A&=&kB^2 \\ (6)&=&k(3)^2 \\ \\ \dfrac{6}{9}&=&\dfrac{k(3)^2}{9} \\ \\ k&=&\dfrac{6}{9}\text{ or }\dfrac{2}{3} \end{array}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ C&=&kAB \\ (24)&=&k(3)(2) \\ \\ \dfrac{24}{6}&=&\dfrac{k(3)(2)}{6} \\ \\ k&=&4 \end{array}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ y&=&\dfrac{k}{x^3} \\ \\ (54)&=&\dfrac{k}{(3)^3} \\ \\ 54&=&\dfrac{k}{27} \\ \\ 27\cdot 54&=&\dfrac{k}{\cancel{27}}\cdot \cancel{27} \\ \\ k&=&1458 \end{array}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ x&=&kY \\ (12)&=&k(8) \\ \\ \dfrac{12}{8}&=&\dfrac{k(8)}{8} \\ \\ k&=&\dfrac{12}{8}\text{ or }\dfrac{3}{2} \end{array}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ \\ A&=&kB^2\sqrt{C} \\ (25)&=&k(5)^2\sqrt{(9)} \\ 25&=&k(75) \\ \\ k&=&\dfrac{25}{75} \\ \\ k&=&\dfrac{1}{3} \end{array}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ \\ \\ y&=&\dfrac{kmn^2}{d} \\ \\ (10)&=&\dfrac{k(4)(5)^2}{(6)} \\ \\ k&=&\dfrac{\cancel{10}\cancel{5}\cdot \cancel{6}3}{\cancel{(4)}(5)^{\cancel{2}}} \\ \\ k&=&\dfrac{3}{5} \end{array}$
$\begin{array}{rrl} \\ \\ \\ \\ \\ \\ \\ \\ P&=&\dfrac{kT}{V} \\ \\ (10)&=&\dfrac{k(250)}{(400)} \\ \\ k&=&\dfrac{10(400)}{250} \\ \\ k&=&16 \end{array}$
$\phantom{1}$
$I=kV \\$
$\begin{array}{ll} \begin{array}{rrl} &&\textbf{1st Data} \\ I&=&5 \text{ A} \\ V&=&15\text{ V} \\ k&=&\text{find} \\ \\ I&=&kV \\ 5\text{ A}&=&k(\text{15 V}) \\ \\ k&=&\dfrac{\text{5 A}}{\text{15 V}} \\ \\ k&=&\dfrac{1}{3}\text{ A/V} \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ I&=&\text{find} \\ k&=&\dfrac{1}{3} \\ \\ V&=&\text{25 V} \\ \\ I&=&kV \\ I&=&\left(\dfrac{1}{3}\right)(25) \\ \\ I&=&8\dfrac{1}{3}\text{ A} \end{array} \end{array}$
$\phantom{1}$
$I=\dfrac{k}{R} \\$
$\begin{array}{ll} \begin{array}{rrl} &&\textbf{1st Data} \\ I&=&\text{12 A} \\ k&=&\text{find} \\ R&=&240\Omega \\ \\ I&=&\dfrac{k}{R} \\ \\ \text{12 A}&=&\dfrac{k}{240\Omega} \\ \\ k&=&(\text{12 A})(240\Omega) \\ k&=&2880\text{ A}\Omega \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ I&=&\text{find} \\ k&=&2880 \\ R&=&540\Omega \\ \\ I&=&\dfrac{k}{R} \\ \\ I&=&\dfrac{2880\text{ A}\Omega}{540\Omega} \\ \\ I&=&5.\bar{3}\text{ A or }5\dfrac{1}{3} \end{array} \end{array}$
$\phantom{1}$
$d_{\text{s}}=km \\$
$\begin{array}{ll} \begin{array}{rrl} \\ \\ \\ \\ &&\textbf{1st Data} \\ d_{\text{s}}&=&18\text{ cm} \\ k&=&\text{find} \\ m&=&3\text{ kg} \\ \\ 18\text{ cm}&=&k(3\text{ kg}) \\ \\ k&=&\dfrac{\text{18 cm}}{\text{3 kg}} \\ \\ k&=&\text{6 cm/kg} \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ d_{\text{s}}&=&\text{find} \\ k&=&\text{6 cm/kg} \\ m&=&\text{5 kg} \\ \\ d_{\text{s}}&=&(\text{6 cm/kg})(\text{5 kg}) \\ d_{\text{s}}&=&\text{30 cm} \end{array} \end{array}$
$\phantom{1}$
$V=\dfrac{k}{P} \\$
$\begin{array}{ll} \begin{array}{rrl} \\ &&\textbf{1st Data} \\ P&=&32\text{ kg/cm}^2 \\ V&=&200\text{ cm}^3 \\ k&=&\text{find} \\ \\ 200\text{ cm}^3&=&\dfrac{k}{32\text{ kg/cm}^2} \\ \\ k&=&(200\text{ cm}^3)(32\text{ kg/cm}^2) \\ k&=&6400\text{ kg cm} \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ P&=&40 \\ V&=&\text{find} \\ k&=&6400 \\ \\ V&=&\dfrac{6400}{40} \\ \\ V&=&160\text{ cm}^3 \end{array} \end{array}$
$\phantom{1}$
$c=kP \\$
$\begin{array}{ll} \begin{array}{rrl} \\ \\ \\ &&\textbf{1st Data} \\ c&=&60,000 \\ k&=&\text{find} \\ P&=&250 \\ \\ 60,000&=&k(250) \\ \\ k&=&\dfrac{60,000}{250} \\ \\ k&=&240 \end{array} & \hspace{0.5in} \begin{array}{rrl} && \textbf{2nd Data} \\ c&=&\text{find} \\ k&=&240 \\ P&=&1,000,000 \\ \\ c&=&(240)(1,000,000) \\ c&=&240,000,000\text{ or 240 million} \end{array} \end{array}$
$\phantom{1}$
$t=\dfrac{k}{b} \\$
$\begin{array}{ll} \begin{array}{rrl} \\ &&\textbf{1st Data} \\ t&=&5\text{ h} \\ k&=&\text{find} \\ b&=&7 \\ \\ 5\text{ h}&=&\dfrac{k}{7} \\ \\ k&=&\text{(5 h)}(7) \\ k&=&35 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ t&=&\text{find} \\ k&=&35 \\ b&=&10 \\ \\ t&=&\dfrac{35}{10} \\ \\ t&=&3.5\text{ h} \end{array} \end{array}$
$\phantom{1}$
$\lambda=\dfrac{k}{f} \\$
$\begin{array}{ll} \begin{array}{rrl} \\ &&\textbf{1st Data} \\ \lambda&=&250\text{ m} \\ k&=&\text{find} \\ f&=&1200\text{ kHz} \\ \\ 250&=&\dfrac{k}{1200} \\ \\ k&=&(250)(1200) \\ k&=&300,000 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ \lambda&=&\text{find} \\ k&=&300,000 \\ f&=&60\text{ kHz} \\ \\ \lambda&=&\dfrac{300,000}{60} \\ \\ \lambda&=&5000\text{ m} \end{array} \end{array}$
$\phantom{1}$
$w=km \\$
$\begin{array}{ll} \begin{array}{rrl} \\ && \textbf{1st Data} \\ w&=&64\text{ kg} \\ k&=&\text{find} \\ m&=&96\text{ kg} \\ \\ 64&=&k(96) \\ \\ k&=&\dfrac{64}{96} \\ \\ k&=&\dfrac{2}{3} \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ w&=&\text{find} \\ k&=&\dfrac{2}{3} \\ m&=&60\text{ kg} \\ \\ w&=&\left(\dfrac{2}{3}\right)(60\text{ kg}) \\ \\ w&=&40\text{ kg} \end{array} \end{array}$
$\phantom{1}$
$t=\dfrac{d}{v} \\$
$\begin{array}{ll} \begin{array}{rrl} &&\textbf{1st Data} \\ t&=&\text{5 h} \\ d&=&\text{find} \\ v&=&\text{80 km/h} \\ \\ \text{5 h}&=&\dfrac{d}{\text{80 km/h}} \\ \\ d&=&5(80) \\ d&=&\text{400 km} \end{array} & \hspace{0.5in} \begin{array}{rrl} \\ \\ &&\textbf{2nd Data} \\ t&=&\text{4.2 h} \\ d&=&\text{400 km} \\ v&=&\text{find} \\ \\ 4.2&=&\dfrac{400}{v} \\ \\ v&=&\dfrac{400}{4.2} \\ \\ v&=&95.24\text{ km/h} \end{array} \end{array}$
$\phantom{1}$
$V=khr^2 \\$
$\begin{array}{ll} \begin{array}{rrl} \\ \\ \\ &&\textbf{1st Data} \\ V&=&33.5\text{ cm}^3 \\ k&=&\text{find} \\ h&=&\text{8 cm} \\ r&=&\text{2 cm} \\ \\ 33.5&=&k(8)(2)^2 \\ \\ k&=&\dfrac{33.5}{(8)(2)^2} \\ \\ k&=&1.046875 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ V&=&\text{find} \\ k&=&1.046875 \\ h&=&\text{6 cm} \\ r&=&\text{4 cm} \\ \\ V&=&khr^2 \\ V&=&(1.046875)(6)(4)^2 \\ V&=&100.5\text{ cm}^3 \end{array} \end{array}$
$\phantom{1}$
$F_{\text{e}}=\dfrac{kv^2}{r} \\$
$\begin{array}{ll} \begin{array}{rrl} \\ \\ \\ \\ &&\textbf{1st Data} \\ F_{\text{e}}&=&100\text{ N} \\ k&=&\text{find} \\ v&=&10\text{ m/s} \\ r&=&\text{0.5 m} \\ \\ 100\text{ N}&=&\dfrac{k(10 \text{ m/s})^2}{\text{0.5 m}} \\ \\ k&=&\dfrac{(0.5)(100)}{(10)^2} \\ \\ k&=&0.5 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ F_{\text{e}}&=&\text{find} \\ k&=&0.5 \\ v&=&25\text{ m/s} \\ r&=&1.0\text{ m} \\ \\ F_{\text{e}}&=&\dfrac{0.5(25)^2}{1.0} \\ \\ F_{\text{e}}&=&312.5\text{ N} \end{array} \end{array}$
$\phantom{1}$
$L_{\text{max}}=\dfrac{kd^4}{h^2} \\$
$\begin{array}{ll} \begin{array}{rrl} \\ \\ \\ &&\textbf{1st Data} \\ L_{\text{max}}&=&64\text{ tonnes} \\ k&=&\text{find} \\ d&=&2.0\text{ m} \\ h&=&8.0\text{ m} \\ \\ 64&=&\dfrac{k(2)^4}{(8)^2} \\ \\ k&=&\dfrac{64(8)^2}{(2)^4} \\ \\ k&=&256 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ L_{\text{max}}&=&\text{find} \\ k&=&256 \\ d&=&3.0\text{ m} \\ h&=&12.0\text{ m} \\ \\ L_{\text{max}}&=&\dfrac{(256)(3.0)^4}{(12.0)^2} \\ \\ L_{\text{max}}&=&144\text{ tonnes} \end{array} \end{array}$
$\phantom{1}$
$V=\dfrac{kT}{P} \\$
$\begin{array}{ll} \begin{array}{rrl} \\ \\ \\ \\ \\ &&\textbf{1st Data} \\ V&=&225\text{ cc} \\ k&=&\text{find} \\ T&=&300\text{ K} \\ P&=&100\text{ N/cm}^2 \\ \\ V&=&\dfrac{kT}{P} \\ \\ 225&=&\dfrac{k(300)}{100} \\ \\ k&=&\dfrac{225(100)}{300} \\ \\ k&=&75 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ V&=&\text{find} \\ k&=&75 \\ T&=&270 \\ P&=&150 \\ \\ V&=&\dfrac{75(270)}{150} \\ \\ V&=&135\text{ cc} \end{array} \end{array}$
$\phantom{1}$
$R=\dfrac{kl}{d^2} \\$
$\begin{array}{ll} \begin{array}{rrl} \\ \\ \\ \\ \\ &&\textbf{1st Data} \\ R&=&20\Omega \\ k&=&\text{find} \\ l&=&5.0\text{ m} \\ d&=&0.25\text{ cm} \\ \\ R&=&\dfrac{kl}{d^2} \\ \\ 20\Omega&=&\dfrac{k(5.0\text{ m})}{\text{(0.25 cm)}^2} \\ \\ k&=&\dfrac{(20 \Omega)\text{(0.25 cm)}^2}{\text{5.0 m}} \\ \\ k&=&0.25 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ R&=&\text{find} \\ k&=&0.25 \\ l&=&10.0\text{ m} \\ d&=&0.50\text{ cm} \\ \\ R&=&\dfrac{(0.25)\text{(10.0 m)}}{\text{(0.50 cm)}^2} \\ \\ R&=&10\Omega \end{array} \end{array}$
$\phantom{1}$
$V=khd^2 \\$
$\begin{array}{ll} \begin{array}{rrl} &&\textbf{1st Data} \\ V&=&377\text{ m}^3 \\ k&=&\text{find} \\ h&=&30\text{ m} \\ d&=&2.0\text{ m} \\ \\ 377\text{ m}^3&=&k(30)(2.0)^2 \\ \\ k&=&\dfrac{377}{(30)(2.0)^2} \\ \\ k&=&3.1416 \end{array} & \hspace{0.5in} \begin{array}{rrl} &&\textbf{2nd Data} \\ V&=&225\text{ m}^3 \\ k&=&3.1416 \\ h&=&\text{find} \\ d&=&1.75\text{ m} \\ \\ 225&=&\pi h(1.75)^2 \\ \\ h&=&\dfrac{225}{\pi (1.75)^2} \\ \\ h&=&23.4\text{ m} \end{array} \end{array}$

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