Chapter 1, Problem 1.32P

(a)

Interpretation Introduction

Interpretation:

Correct number of significant figures has to be given for $z$ in the calculation $213.3642+17.54+32.978=z$.

Concept Introduction:

Precision of the quantities that are measured use the significant figures to express the result.  There are six rules for determining the number of significant figures.  They are given as follows.

• Nonzero digits and the zeros that are present between nonzero digits are significant.
• Zeros that are used merely to position the decimal are not significant.
• If zeros are present in the right end of the decimal point, then the zeros are significant.
• If zeros are present in the result are not in right of the decimal point, then the zeros may or may not be significant.
• If the zeros disappear when scientific notation is used means then those zeros are not significant.
• Exact numbers have no limit to their precision and they do not obey the rules given for significant figures.

Answer for a calculation should be always expressed in least significant figures in case of multiplication and division.

In case of addition and subtraction, the answer should have least significant figures after the decimal point.

Explanation of Solution

Given calculation is $213.3642+17.54+32978=z$.  Number of digits that is present after the decimal point in $213.3642$ is four.  Number of digits present after decimal point in $17.54$ is two.  Number of digits present after decimal point in $32978$ is zero.  As it is an addition, the answer should contain only two digits after decimal point.

  $\begin{array}{c}213.3642+17.54+32978=z\\ \text{33208}\text{.9042}=z\\ 33209=z\end{array}$

Rounding off to the correct number of significant figures, the value of $z$ is $33209$.

(b)

Interpretation Introduction

Interpretation:

Correct number of significant figures has to be given for $z$ in the calculation $373.26-119=z$.

Concept Introduction:

Refer part (a).

Explanation of Solution

Given calculation is $373.26-119=z$.  Number of digits that is present after the decimal point in $373.26$ is two.  Number of digits present after decimal point in $119$ is zero.  As it is a subtraction, the answer should not contain any digit after decimal point.

  $\begin{array}{c}373.26-119=z\\ 254.26=z\\ 254=z\end{array}$

Rounding off to the correct number of significant figures, the value of $z$ is $254$.

(c)

Interpretation Introduction

Interpretation:

Correct number of significant figures has to be given for $z$ in the calculation $\text{(6}\text{.626075}\times {\text{10}}^{\text{-34}})(2.997925\times {10}^8)/(1.38066\times {10}^{-23})=z$.

Concept Introduction:

Refer part (a).

Explanation of Solution

Given calculation is $\text{(6}\text{.626075}\times {\text{10}}^{\text{-34}})(2.997925\times {10}^8)/(1.38066\times {10}^{-23})=z$.  Number of significant figures in $\text{6}\text{.626075}\times {\text{10}}^{\text{-34}}$ is seven, in $2.997925\times {10}^8$ is seven, and in $1.38066\times {10}^{-23}$ is six.  As it is multiplication and division, the answer should contain least significant figure and that is six significant figures.

  $\begin{array}{c}\frac{\text{(6}\text{.626075}\times {\text{10}}^{\text{-34}})(2.997925\times {10}^8)}{(1.38066\times {10}^{-23})}=z\\ \frac{\text{19}\text{.864475894375}\times {10}^{-26}}{1.38066\times {10}^{-23}}=z\\ \text{14}\text{.38766669156418}\times {10}^{-3}=z\\ 14.3877\times {10}^{-3}=z\\ 0.0143877=z\end{array}$

Rounding off to the correct number of significant figures, the value of $z$ is $0.0143877$.

(d)

Interpretation Introduction

Interpretation:

Correct number of significant figures has to be given for $z$ in the calculation $(9.109390\times {10}^{-31}+1.67262\times {10}^{-27}-1.67493\times {10}^{-27})(2.997925\times {10}^8)=z$.

Concept Introduction:

Refer part (a).

Explanation of Solution

Given calculation is $(9.109390\times {10}^{-31}+1.67262\times {10}^{-27}-1.67493\times {10}^{-27})(2.997925\times {10}^8)=z$.  Number of significant figures in $9.109390$ is seven, in $1.67262$ is six, in $1.67493$ is six, and in $2.997925$ is seven.  As it is multiplication, the answer should contain least significant figure that is six significant figures.

  $\begin{array}{c}(9.109390\times {10}^{-31}+1.67262\times {10}^{-27}-1.67493\times {10}^{-27}){(2.997925\times {10}^8)}^2=z\\ (0.0009109390\times {10}^{-27}+1.67292\times {10}^{-27}-1.67493\times {10}^{-27}){(2.997925\times {10}^8)}^2=z\\ \text{(-0}\text{.001099061}\times {10}^{-27}){(2.997925\times {10}^8)}^2=z\\ \text{-0}\text{.0098778704226945}\times {10}^{-11}=z\\ -9.87789\times {10}^{-8}=z\end{array}$

Rounding off to the correct number of significant figures, the value of $z$ is $-9.87789\times {10}^{-8}$.