Answer Key 2.7
- \(x=ky\)
- \(x=kyz\)
- \(x=\dfrac{k}{y}\)
- \(x=ky^2\)
- \(x=kzy\)
- \(x=\dfrac{k}{y^3}\)
- \(x=ky^2\sqrt{z}\)
- \(x=\dfrac{k}{y^6}\)
- \(x=\dfrac{ky^3}{\sqrt{z}}\)
- \(x=\dfrac{k}{y^2\sqrt{z}}\)
- \(x=\dfrac{kzy}{p^3}\)
- \(x=\dfrac{k}{y^3z^2}\)
- \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\
A&=&kB \\
(15)&=&k(5) \\ \\
\dfrac{15}{5}&=&\dfrac{k(5)}{5} \\ \\
k&=&3
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\
P&=&kQR \\
(12)&=&k(8)(3) \\ \\
\dfrac{12}{24}&=&\dfrac{k(8)(3)}{24} \\ \\
k&=&\dfrac{1}{2}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\
A&=&\dfrac{k}{B} \\ \\
(7)&=&\dfrac{k}{(4)} \\ \\
(4)7&=&\dfrac{k}{\cancel{4}}\cancel{(4)} \\ \\
k&=&28
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\
A&=&kB^2 \\
(6)&=&k(3)^2 \\ \\
\dfrac{6}{9}&=&\dfrac{k(3)^2}{9} \\ \\
k&=&\dfrac{6}{9}\text{ or }\dfrac{2}{3}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\
C&=&kAB \\
(24)&=&k(3)(2) \\ \\
\dfrac{24}{6}&=&\dfrac{k(3)(2)}{6} \\ \\
k&=&4
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{k}{x^3} \\ \\
(54)&=&\dfrac{k}{(3)^3} \\ \\
54&=&\dfrac{k}{27} \\ \\
27\cdot 54&=&\dfrac{k}{\cancel{27}}\cdot \cancel{27} \\ \\
k&=&1458
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\
x&=&kY \\
(12)&=&k(8) \\ \\
\dfrac{12}{8}&=&\dfrac{k(8)}{8} \\ \\
k&=&\dfrac{12}{8}\text{ or }\dfrac{3}{2}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\
A&=&kB^2\sqrt{C} \\
(25)&=&k(5)^2\sqrt{(9)} \\
25&=&k(75) \\ \\
k&=&\dfrac{25}{75} \\ \\
k&=&\dfrac{1}{3}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\
y&=&\dfrac{kmn^2}{d} \\ \\
(10)&=&\dfrac{k(4)(5)^2}{(6)} \\ \\
k&=&\dfrac{\cancel{10}\cancel{5}\cdot \cancel{6}3}{\cancel{(4)}(5)^{\cancel{2}}} \\ \\
k&=&\dfrac{3}{5}
\end{array}\) - \(\begin{array}{rrl}
\\ \\ \\ \\ \\ \\ \\ \\
P&=&\dfrac{kT}{V} \\ \\
(10)&=&\dfrac{k(250)}{(400)} \\ \\
k&=&\dfrac{10(400)}{250} \\ \\
k&=&16
\end{array}\) - \(\phantom{1}\)
\(I=kV \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
&&\textbf{1st Data} \\
I&=&5 \text{ A} \\
V&=&15\text{ V} \\
k&=&\text{find} \\ \\
I&=&kV \\
5\text{ A}&=&k(\text{15 V}) \\ \\
k&=&\dfrac{\text{5 A}}{\text{15 V}} \\ \\
k&=&\dfrac{1}{3}\text{ A/V}
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
I&=&\text{find} \\
k&=&\dfrac{1}{3} \\ \\
V&=&\text{25 V} \\ \\
I&=&kV \\
I&=&\left(\dfrac{1}{3}\right)(25) \\ \\
I&=&8\dfrac{1}{3}\text{ A}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(I=\dfrac{k}{R} \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
&&\textbf{1st Data} \\
I&=&\text{12 A} \\
k&=&\text{find} \\
R&=&240\Omega \\ \\
I&=&\dfrac{k}{R} \\ \\
\text{12 A}&=&\dfrac{k}{240\Omega} \\ \\
k&=&(\text{12 A})(240\Omega) \\
k&=&2880\text{ A}\Omega
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
I&=&\text{find} \\
k&=&2880 \\
R&=&540\Omega \\ \\
I&=&\dfrac{k}{R} \\ \\
I&=&\dfrac{2880\text{ A}\Omega}{540\Omega} \\ \\
I&=&5.\bar{3}\text{ A or }5\dfrac{1}{3}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(d_{\text{s}}=km \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\ \\ \\ \\
&&\textbf{1st Data} \\
d_{\text{s}}&=&18\text{ cm} \\
k&=&\text{find} \\
m&=&3\text{ kg} \\ \\
18\text{ cm}&=&k(3\text{ kg}) \\ \\
k&=&\dfrac{\text{18 cm}}{\text{3 kg}} \\ \\
k&=&\text{6 cm/kg}
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
d_{\text{s}}&=&\text{find} \\
k&=&\text{6 cm/kg} \\
m&=&\text{5 kg} \\ \\
d_{\text{s}}&=&(\text{6 cm/kg})(\text{5 kg}) \\
d_{\text{s}}&=&\text{30 cm}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(V=\dfrac{k}{P} \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\
&&\textbf{1st Data} \\
P&=&32\text{ kg/cm}^2 \\
V&=&200\text{ cm}^3 \\
k&=&\text{find} \\ \\
200\text{ cm}^3&=&\dfrac{k}{32\text{ kg/cm}^2} \\ \\
k&=&(200\text{ cm}^3)(32\text{ kg/cm}^2) \\
k&=&6400\text{ kg cm}
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
P&=&40 \\
V&=&\text{find} \\
k&=&6400 \\ \\
V&=&\dfrac{6400}{40} \\ \\
V&=&160\text{ cm}^3
\end{array}
\end{array}\) - \(\phantom{1}\)
\(c=kP \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\ \\ \\
&&\textbf{1st Data} \\
c&=&60,000 \\
k&=&\text{find} \\
P&=&250 \\ \\
60,000&=&k(250) \\ \\
k&=&\dfrac{60,000}{250} \\ \\
k&=&240
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&& \textbf{2nd Data} \\
c&=&\text{find} \\
k&=&240 \\
P&=&1,000,000 \\ \\
c&=&(240)(1,000,000) \\
c&=&240,000,000\text{ or 240 million}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(t=\dfrac{k}{b} \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\
&&\textbf{1st Data} \\
t&=&5\text{ h} \\
k&=&\text{find} \\
b&=&7 \\ \\
5\text{ h}&=&\dfrac{k}{7} \\ \\
k&=&\text{(5 h)}(7) \\
k&=&35
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
t&=&\text{find} \\
k&=&35 \\
b&=&10 \\ \\
t&=&\dfrac{35}{10} \\ \\
t&=&3.5\text{ h}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(\lambda=\dfrac{k}{f} \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\
&&\textbf{1st Data} \\
\lambda&=&250\text{ m} \\
k&=&\text{find} \\
f&=&1200\text{ kHz} \\ \\
250&=&\dfrac{k}{1200} \\ \\
k&=&(250)(1200) \\
k&=&300,000
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
\lambda&=&\text{find} \\
k&=&300,000 \\
f&=&60\text{ kHz} \\ \\
\lambda&=&\dfrac{300,000}{60} \\ \\
\lambda&=&5000\text{ m}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(w=km \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\
&& \textbf{1st Data} \\
w&=&64\text{ kg} \\
k&=&\text{find} \\
m&=&96\text{ kg} \\ \\
64&=&k(96) \\ \\
k&=&\dfrac{64}{96} \\ \\
k&=&\dfrac{2}{3}
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
w&=&\text{find} \\
k&=&\dfrac{2}{3} \\
m&=&60\text{ kg} \\ \\
w&=&\left(\dfrac{2}{3}\right)(60\text{ kg}) \\ \\
w&=&40\text{ kg}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(t=\dfrac{d}{v} \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
&&\textbf{1st Data} \\
t&=&\text{5 h} \\
d&=&\text{find} \\
v&=&\text{80 km/h} \\ \\
\text{5 h}&=&\dfrac{d}{\text{80 km/h}} \\ \\
d&=&5(80) \\
d&=&\text{400 km}
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
\\ \\
&&\textbf{2nd Data} \\
t&=&\text{4.2 h} \\
d&=&\text{400 km} \\
v&=&\text{find} \\ \\
4.2&=&\dfrac{400}{v} \\ \\
v&=&\dfrac{400}{4.2} \\ \\
v&=&95.24\text{ km/h}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(V=khr^2 \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\ \\ \\
&&\textbf{1st Data} \\
V&=&33.5\text{ cm}^3 \\
k&=&\text{find} \\
h&=&\text{8 cm} \\
r&=&\text{2 cm} \\ \\
33.5&=&k(8)(2)^2 \\ \\
k&=&\dfrac{33.5}{(8)(2)^2} \\ \\
k&=&1.046875
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
V&=&\text{find} \\
k&=&1.046875 \\
h&=&\text{6 cm} \\
r&=&\text{4 cm} \\ \\
V&=&khr^2 \\
V&=&(1.046875)(6)(4)^2 \\
V&=&100.5\text{ cm}^3
\end{array}
\end{array}\) - \(\phantom{1}\)
\(F_{\text{e}}=\dfrac{kv^2}{r} \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\ \\ \\ \\
&&\textbf{1st Data} \\
F_{\text{e}}&=&100\text{ N} \\
k&=&\text{find} \\
v&=&10\text{ m/s} \\
r&=&\text{0.5 m} \\ \\
100\text{ N}&=&\dfrac{k(10 \text{ m/s})^2}{\text{0.5 m}} \\ \\
k&=&\dfrac{(0.5)(100)}{(10)^2} \\ \\
k&=&0.5
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
F_{\text{e}}&=&\text{find} \\
k&=&0.5 \\
v&=&25\text{ m/s} \\
r&=&1.0\text{ m} \\ \\
F_{\text{e}}&=&\dfrac{0.5(25)^2}{1.0} \\ \\
F_{\text{e}}&=&312.5\text{ N}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(L_{\text{max}}=\dfrac{kd^4}{h^2} \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\ \\ \\
&&\textbf{1st Data} \\
L_{\text{max}}&=&64\text{ tonnes} \\
k&=&\text{find} \\
d&=&2.0\text{ m} \\
h&=&8.0\text{ m} \\ \\
64&=&\dfrac{k(2)^4}{(8)^2} \\ \\
k&=&\dfrac{64(8)^2}{(2)^4} \\ \\
k&=&256
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
L_{\text{max}}&=&\text{find} \\
k&=&256 \\
d&=&3.0\text{ m} \\
h&=&12.0\text{ m} \\ \\
L_{\text{max}}&=&\dfrac{(256)(3.0)^4}{(12.0)^2} \\ \\
L_{\text{max}}&=&144\text{ tonnes}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(V=\dfrac{kT}{P} \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\ \\ \\ \\ \\
&&\textbf{1st Data} \\
V&=&225\text{ cc} \\
k&=&\text{find} \\
T&=&300\text{ K} \\
P&=&100\text{ N/cm}^2 \\ \\
V&=&\dfrac{kT}{P} \\ \\
225&=&\dfrac{k(300)}{100} \\ \\
k&=&\dfrac{225(100)}{300} \\ \\
k&=&75
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
V&=&\text{find} \\
k&=&75 \\
T&=&270 \\
P&=&150 \\ \\
V&=&\dfrac{75(270)}{150} \\ \\
V&=&135\text{ cc}
\end{array}
\end{array}\) - \(\phantom{1}\)
\(R=\dfrac{kl}{d^2} \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
\\ \\ \\ \\ \\
&&\textbf{1st Data} \\
R&=&20\Omega \\
k&=&\text{find} \\
l&=&5.0\text{ m} \\
d&=&0.25\text{ cm} \\ \\
R&=&\dfrac{kl}{d^2} \\ \\
20\Omega&=&\dfrac{k(5.0\text{ m})}{\text{(0.25 cm)}^2} \\ \\
k&=&\dfrac{(20 \Omega)\text{(0.25 cm)}^2}{\text{5.0 m}} \\ \\
k&=&0.25
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
R&=&\text{find} \\
k&=&0.25 \\
l&=&10.0\text{ m} \\
d&=&0.50\text{ cm} \\ \\
R&=&\dfrac{(0.25)\text{(10.0 m)}}{\text{(0.50 cm)}^2} \\ \\
R&=&10\Omega
\end{array}
\end{array}\) - \(\phantom{1}\)
\(V=khd^2 \\ \)
\(\begin{array}{ll}
\begin{array}{rrl}
&&\textbf{1st Data} \\
V&=&377\text{ m}^3 \\
k&=&\text{find} \\
h&=&30\text{ m} \\
d&=&2.0\text{ m} \\ \\
377\text{ m}^3&=&k(30)(2.0)^2 \\ \\
k&=&\dfrac{377}{(30)(2.0)^2} \\ \\
k&=&3.1416
\end{array}
& \hspace{0.5in}
\begin{array}{rrl}
&&\textbf{2nd Data} \\
V&=&225\text{ m}^3 \\
k&=&3.1416 \\
h&=&\text{find} \\
d&=&1.75\text{ m} \\ \\
225&=&\pi h(1.75)^2 \\ \\
h&=&\dfrac{225}{\pi (1.75)^2} \\ \\
h&=&23.4\text{ m}
\end{array}
\end{array}\)
