Chapter 2, Problem 2.4P

To determine

The dimensions value from $A$ to $CC$.

Answer to Problem 2.4P

The dimension for $A$ is $0.1^{″}$, the dimension for $B$ is $1.2^{″}$, the dimension for $C$ is $1.6^{″}$, the dimension for $D$ is $23^{\prime }$, the dimension for $E$ is $285^{\prime }$, the dimension for $F$ is $0.1^{″}$, the dimension for $G$ is $2^{\prime }$, the dimension for $H$ is $35^{\prime }$, the dimension for $I$ is $570^{\prime }$, the dimension for $J$ is $0.1^{\prime }$, the dimension for $K$ is $3^{\prime }$, the dimension for $L$ is $45^{\prime }$, the dimension for $M$ is $550^{\prime }$, the dimension for $N$ is $6900^{\prime }$, the dimension for $O$ is $0.1^{″}$, the dimension for $P$ is $4^{\prime }$, the dimension for $Q$ is $53^{\prime }$, the dimension for $R$ is $660^{\prime }$ V, the dimension for $S$ is $7400^{\prime }$, the dimension for $T$ is $0.1^{″}$, the dimension for $U$ is $5^{\prime }$, the dimension for $V$ is $64^{\prime }$, the dimension for $W$ is $770^{\prime }$, the dimension for $X$ is $8300^{\prime }$, the dimension for $Y$ is $0.1^{″}$, the dimension for $Z$ is $1^{″}$, the dimension for $AA$ is $89^{\prime }$, the dimension for $BB$ is $1150^{\prime }$ and the dimension for $CC$ is $13750^{\prime }$.

Explanation of Solution

Given:

The dimensions for $10$, $20$, $30$, $40$, $50$ and $60$ scales are given.

The ratio for $A$ is $1:1$.

The ratio for $B$ is $1:1$.

The ratio for $C$ is $1^{″}=1^{\prime }$.

The ratio for $D$ is $1^{″}=10^{\prime }$.

The ratio for $E$ is $1^{″}=100^{\prime }$.

The ratio for $F$ is $1:2$.

The ratio for $G$ is $1^{\prime }=2^{″}$.

The ratio for $H$ is $1^{″}=20^{\prime }$.

The ratio for $I$ is $1^{″}=200^{\prime }$.

The ratio for $J$ is $1:3$.

The ratio for $K$ is $1^{″}=3^{\prime }$.

The ratio for $L$ is $1^{″}=30^{\prime }$.

The ratio for $M$ is $1^{″}=300^{\prime }$.

The ratio for $N$ is $1^{″}=3000^{\prime }$.

The ratio for $O$ is $1:4$.

The ratio for $P$ is $1^{″}=4^{\prime }$.

The ratio for $Q$ is $1^{″}=40^{\prime }$.

The ratio for $R$ is $1^{″}=400^{\prime }$.

The ratio for $S$ is $1^{″}=4000^{\prime }$.

The ratio for $T$ is $1:5$.

The ratio for $U$ is $1^{″}=5^{\prime }$.

The ratio for $V$ is $1^{″}=50^{\prime }$.

The ratio for $W$ is $1^{″}=500^{\prime }$.

The ratio for $X$ is $1^{″}=5000^{\prime }$.

The ratio for $Y$ is $1:6$.

The ratio for $Z$ is $1^{″}=6^{\prime }$.

The ratio for $AA$ is $1^{″}=60^{\prime }$.

The ratio for $BB$ is $1^{″}=600^{\prime }$.

The ratio for $CC$ is $1^{″}=6000^{\prime }$

Calculation:

The dimension of $10$ scales are calculated as below,

Here $1^{″}=10\text{divisions}$

The dimension of $A$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }A=\frac{1^{″}}{10}\\ =0.1^{″}\end{array}$

The dimension of $B$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }B=1^{″}+2\text{divisions}\\ =1.2^{″}\end{array}$

The dimension of $C$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }C=1^{″}+6\text{divisions}\\ =1.6^{″}\end{array}$

The dimension of $D$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }D\text{ in}\text{}\text{inches}=2^{″}+3\text{divisions}\\ =2.3^{″}\end{array}$

The ratio for $D$ is $1^{″}=10^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }D\text{in}\text{}\text{feets}=2.3^{″}\times 10\\ =23^{\prime }\end{array}$

The dimension for $E$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }E\text{in}\text{}\text{inches}=2^{″}+8.5\text{divisions}\\ =2.85^{″}\end{array}$

The ratio for $E$ is $1^{″}=100^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }E\text{in}\text{}\text{feets}=2.85^{″}\times 100\\ =285^{\prime }\end{array}$

The dimension of $20$ scale is calculated as below

Here, $1^{″}=20\text{divisions}$

The dimension of $F$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }F=\left(\frac{1^{″}}{20}\right)\times 2\\ =\left(0.05\right)\times 2\\ =0.1^{″}\end{array}$

The dimension of $G$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }G\text{in}\text{}\text{inches}=20\text{divisions}\\ =1^{″}\end{array}$

The ratio for $G$ is $1^{\prime }=2^{″}$.

$\begin{array}{c}\text{dimension}\text{of }G\text{ in}\text{}\text{feets}=1^{″}\times 2\\ =2^{\prime }\end{array}$

The dimension of $H$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }H\text{in}\text{}\text{inches}=35\text{divisions}\\ ={\left(\frac{35}{20}\right)}^{\prime \text{}\prime }\\ =1.75^{″}\end{array}$

The ratio for $H$ is $1^{″}=20^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }H\text{ in}\text{}\text{feets}=1.75^{″}\times 20\\ =35^{\prime }\end{array}$

The dimension of $I$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }I\text{in}\text{}\text{inches}=57\text{divisions}\\ ={\left(\frac{57}{20}\right)}^{\prime \text{}\prime }\\ =2.85^{″}\end{array}$

The ratio for $I$ is $1^{″}=200^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }I\text{ in}\text{}\text{feets}=2.85^{″}\times 200\\ =570^{\prime }\end{array}$

The dimension of $30$ scale is calculated as below

Here, $1^{″}=30\text{divisions}$

The dimension of $J$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }J=\left(\frac{1^{″}}{30}\right)\times 3\\ =\left(0.03\right)\times 3\\ =0.1^{″}\end{array}$

The dimension of $K$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }K\text{in}\text{}\text{inches}=30\text{divisions}\\ ={\left(\frac{30}{30}\right)}^{\prime \text{}\prime }\\ =1^{″}\end{array}$

The ratio for $K$ is $1^{″}=3^{\prime }$

$\begin{array}{c}\text{dimension}\text{of }K\text{ in}\text{}\text{feets}=1^{″}\times 3\\ =3^{\prime }\end{array}$

The dimension of $L$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }L\text{in}\text{}\text{inches}=45\text{divisions}\\ ={\left(\frac{45}{30}\right)}^{\prime \text{}\prime }\\ =1.5^{″}\end{array}$

The ratio for $L$ is $1^{″}=30^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }L\text{ in}\text{}\text{feets}=1.5^{″}\times 30\\ =45^{\prime }\end{array}$

The dimension for $M$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }M\text{in}\text{}\text{inches}=55\text{divisions}\\ ={\left(\frac{55}{30}\right)}^{\prime \text{}\prime }\\ =1.833^{″}\end{array}$

The ratio for $M$ is $1^{″}=300^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }M\text{ in}\text{}\text{feets}=1.83^{″}\times 300\\ =550^{\prime }\end{array}$

The dimension of $N$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }N\text{in}\text{}\text{inches}=69\text{divisions}\\ ={\left(\frac{69}{30}\right)}^{\prime \text{}\prime }\\ =2.3^{″}\end{array}$

The ratio for $N$ is $1^{″}=3000^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }N\text{ in}\text{}\text{feets}=2.3^{″}\times 3000\\ =6900^{\prime }\end{array}$

The dimension for $40$ scale is calculated as below

Here, $1^{″}=40\text{divisions}$

The dimension of $O$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }O=\left(\frac{1^{″}}{40}\right)\times 4\\ =\left(0.025\right)\times 4\\ =0.1^{″}\end{array}$

The dimension of $P$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }P\text{in}\text{}\text{inches}=40\text{divisions}\\ ={\left(\frac{40}{40}\right)}^{\prime \text{}\prime }\\ =1^{″}\end{array}$

The ratio for $P$ is $1^{″}=4^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }P\text{ in}\text{}\text{feets}=1^{″}\times 4\\ =4^{\prime }\end{array}$

The dimension of $Q$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }Q\text{in}\text{}\text{inches}=53\text{divisions}\\ ={\left(\frac{53}{40}\right)}^{\prime \text{}\prime }\\ =1.325^{″}\end{array}$

The ratio for $Q$ is $1^{″}=40^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }Q\text{ in}\text{}\text{feets}=1.325^{″}\times 40\\ =53^{\prime }\end{array}$

The dimension of $R$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }R\text{in}\text{}\text{inches}=66\text{divisions}\\ ={\left(\frac{63}{40}\right)}^{\prime \text{}\prime }\\ =1.65^{″}\end{array}$

The ratio for $R$ is $1^{″}=400^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }R\text{ in}\text{}\text{feets}=1.65^{″}\times 400\\ =660^{\prime }\end{array}$

The dimension of $S$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }S\text{in}\text{}\text{inches}=74\text{divisions}\\ ={\left(\frac{74}{40}\right)}^{\prime \text{}\prime }\\ =1.85^{″}\end{array}$

The ratio for $S$ is $1^{″}=4000^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }S\text{ in}\text{}\text{feets}=1.85^{″}\times 4000\\ =7400^{\prime }\end{array}$

The dimension for $50$ scale is calculated as below

Here, $1^{″}=50\text{divisions}$

The dimension of $T$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }T=\left(\frac{1^{″}}{50}\right)\times 5\\ =\left(0.02\right)\times 5\\ =0.1^{″}\end{array}$

The dimension of $U$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }U\text{in}\text{}\text{inches}=50\text{divisions}\\ ={\left(\frac{50}{50}\right)}^{\prime \text{}\prime }\\ =1^{″}\end{array}$

The ratio for $U$ is $1^{″}=5^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }U\text{ in}\text{}\text{feets}=1^{″}\times 5\\ =5^{\prime }\end{array}$

The dimension for $V$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }V\text{in}\text{}\text{inches}=64\text{divisions}\\ ={\left(\frac{64}{50}\right)}^{\prime \text{}\prime }\\ =1.28^{″}\end{array}$

The ratio for $V$ is $1^{″}=50^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }V\text{ in}\text{}\text{feets}=1.28^{″}\times 50\\ =64^{\prime }\end{array}$

The dimension of $W$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }W\text{in}\text{}\text{inches}=77\text{divisions}\\ ={\left(\frac{77}{50}\right)}^{\prime \text{}\prime }\\ =1.54^{″}\end{array}$

The ratio for $W$ is $1^{″}=500^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }W\text{ in}\text{}\text{feets}=1.54^{″}\times 500\\ =770^{\prime }\end{array}$

The dimension of $X$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }X\text{in}\text{}\text{inches}=83\text{divisions}\\ ={\left(\frac{83}{50}\right)}^{\prime \text{}\prime }\\ =1.66^{″}\end{array}$

The ratio for $X$ is $1^{″}=5000^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }X\text{ in}\text{}\text{feets}=1.66^{″}\times 5000\\ =8300^{\prime }\end{array}$

The dimensions for 60 scale is calculated below

Here, $1^{″}=60\text{divisions}$

The dimension of $Y$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }Y=\left(\frac{1^{″}}{60}\right)\times 6\\ =\left(0.0166\right)\times 6\\ =0.1^{″}\end{array}$

The dimension of $Z$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }Z\text{in}\text{}\text{inches}=60\text{divisions}\\ ={\left(\frac{60}{60}\right)}^{\prime \text{}\prime }\\ =1^{″}\end{array}$

The ratio for $Z$ is $1^{″}=6^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }Z\text{ in}\text{}\text{feets}=1^{″}\times 6\\ =6^{\prime }\end{array}$

The dimension of $AA$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }AA\text{in}\text{}\text{inches}=89\text{divisions}\\ ={\left(\frac{89}{60}\right)}^{\prime \text{}\prime }\\ =1.483^{″}\end{array}$

The ratio for $AA$ is $1^{″}=60^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }AA\text{ in}\text{}\text{feets}=1.483^{″}\times 60\\ =89^{\prime }\end{array}$

The dimension of $BB$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }BB\text{in}\text{}\text{inches}=115\text{divisions}\\ ={\left(\frac{115}{60}\right)}^{\prime \text{}\prime }\\ =1.916^{″}\end{array}$

The ratio for $BB$ is $1^{″}=600^{\prime }$.

$\begin{array}{c}\text{dimension}\text{of }BB\text{ in}\text{}\text{feets}=1.916^{″}\times 600\\ =1150^{\prime }\end{array}$

The dimension $CC$ is calculated as,

$\begin{array}{c}\text{dimension}\text{of }CC\text{in}\text{}\text{inches}=137.5\text{divisions}\\ ={\left(\frac{137.5}{60}\right)}^{\prime \text{}\prime }\\ =2.2916^{″}\end{array}$

The ratio for $CC$ is $1^{″}=6000^{\prime }$

$\begin{array}{c}\text{dimension}\text{of }CC\text{ in}\text{}\text{feets}=2.2916^{″}\times 6000\\ =13750^{\prime }\end{array}$

Conclusion:

Therefore, the dimension for $A$ is $0.1^{″}$, the dimension for $B$ is $1.2^{″}$, the dimension for $C$ is $1.6^{″}$, the dimension for $D$ is $23^{\prime }$, the dimension for $E$ is $285^{\prime }$, the dimension for $F$ is $0.1^{″}$, the dimension for $G$ is $2^{\prime }$, the dimension for $H$ is $35^{\prime }$, the dimension for $I$ is $570^{\prime }$, the dimension for $J$ is $0.1^{\prime }$, the dimension for $K$ is $3^{\prime }$, the dimension for $L$ is $45^{\prime }$, the dimension for $M$ is $550^{\prime }$, the dimension for $N$ is $6900^{\prime }$, the dimension for $O$ is $0.1^{″}$, the dimension for $P$ is $4^{\prime }$, the dimension for $Q$ is $53^{\prime }$, the dimension for $R$ is $660^{\prime }$ V, the dimension for $S$ is $7400^{\prime }$, the dimension for $T$ is $0.1^{″}$, the dimension for $U$ is $5^{\prime }$, the dimension for $V$ is $64^{\prime }$, the dimension for $W$ is $770^{\prime }$, the dimension for $X$ is $8300^{\prime }$, the dimension for $Y$ is $0.1^{″}$, the dimension for $Z$ is $1^{″}$, the dimension for $AA$ is $89^{\prime }$, the dimension for $BB$ is $1150^{\prime }$ and the dimension for $CC$ is $13750^{\prime }$.