Chapter 1, Problem 27QAP

Calculate the following to the correct number of significant figures. Assume that all these numbers are measurements.

(a) $x=17.2+65.18-2.4$(b) $x=\frac{13.0217}{17.10}$(c) $x=\left(0.0061020\right)\left(2.0092\right)\left(1200.00\right)$(d) $x=0.0034+\frac{\sqrt{{\left(0.0034\right)}^2+4\left(1.000\right)\left(6.3\times {10}^{-4}\right)}}{\left(2\right)\left(1.000\right)}$(e) $x=\frac{\left(2.998\times {10}^8\right)\left(3.1\times {10}^{-7}\right)}{6.022\times {10}^{23}}$

Interpretation Introduction

(a)

Interpretation:

The following calculation should be done up to correct significant numbers.

$x=17.2+65.18-2.4$

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after the decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 27QAP

The correct calculation is 80.0.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant number as found in the number with least significant digits in the problem.

As per significant rule 1, the number with least decimal place will be used to decide the significant digits of result.

Firstly, looking at all the three numbers, the least decimal place is up to one.

Hence the calculation result will be made up to correct significant digits as follows:

$x=17.2+65.18-2.4=79.98$

The result has 4 significant digits. To make it to 1 decimal place, we need to round off the nearest number as:

$x=17.2+65.18-2.4=79.98\simeq 80.0$

The calculation result will be 80.0.

Interpretation Introduction

(b)

Interpretation:

The following calculation should be done up to correct significant numbers.

$x=\frac{13.0217}{17.10}$

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 27QAP

The correct calculation is $7.615\times {10}^{-1}$.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant number as found in the number with least significant digits in the problem.

As per significant rule 2, the number with least significant digits will be used to decide the significant digits of result.

Firstly, looking at all the two numbers, the least significant digit is 4 for 17.10.

Hence, the calculation result will be made up to correct significant digits of 4 as follows:

$x=\frac{13.0217}{17.10}=0.7615$

The result has 4 significant digits. Converting in scientific notation as

$x=0.7615=7.615\times {10}^{-1}$

The calculation result will be $7.615\times {10}^{-1}$.

Interpretation Introduction

(c)

Interpretation:

The following calculation should be done up to correct significant numbers.

$x=(0.0061020)(2.0092)(1200.00)$

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 27QAP

The correct calculation is $1.4712\times {10}^1$.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant number as found in the number with least significant digits in the problem.

As per significant rule 2, the number with least significant digit will be used to decide the significant digits of result.

First of all, looking at all the three numbers, the least significant digit is five (2.0092).

Hence the calculation result will be made up to correct significant digits of 5 as follows:

$x=(0.0061020)(2.0092)(1200.00)=14.712$

The result has 5 significant digits. Converting in scientific notation as:

$x=14.712=1.4712\times {10}^1$

The calculation result will be $1.4712\times {10}^1$.

Interpretation Introduction

(d)

Interpretation:

The following calculation should be done up to correct significant numbers.

$x=0.0034+\frac{\sqrt{{(0.0034)}^2+4(1.000)(6.3\times {10}^{-4})}}{(2)(1.000)}$

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after the decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 27QAP

The calculation result will be $x=2.86\times {10}^{-2}$.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant numbers as found in the number with least significant digits in the problem.

As per significant rule 1, the number with least decimal place will be used to decide the significant digits of result.

As per significant rule 2, the number with least significant digit will be used to decide the significant digit of result.

Firstly, the equation should be solved using BODMAS rule simultaneously with round off rule as follows:

The under root terms are multiplied and added as per round off rule (2) and (1) as:

$\begin{array}{c}x=0.0034+\frac{\sqrt{{(0.0034)}^2+4(1.000)(6.3\times {10}^{-4})}}{(2)(1.000)}\\ =0.0034+\frac{\sqrt{0.00001156+0.00252}}{2}\end{array}$

Now, again using rounding off rule (1) for under root addition as:

$\begin{array}{l}x=0.0034+\frac{\sqrt{0.00253156}}{2}\\ x=0.0034+\frac{0.0503146}{2}\\ x=0.0034+0.02516\end{array}$

The result should have two decimal places at least. Hence the result will be obtained by scientific notation as;

$\begin{array}{l}x=0.0034+0.02516=0.0285573\\ x=2.86\times {10}^{-2}\end{array}$

The calculation result will be $x=2.86\times {10}^{-2}$.

Interpretation Introduction

(e)

Interpretation:

The following calculation should be done up to correct significant numbers.

$x=\frac{(2.998\times {10}^8)(3.1\times {10}^{-7})}{6.022\times {10}^{23}}$

Concept introduction:

The significant figures are the numbers which are used to describe a measured quantity. They are calculated with the help of following rules:

1. All of the non-zero digits will be significant.
2. Any zero which is placed between two non-zero digits will be significant.
3. Trailing zeros or if the zero is final, it will be significant only if it is after the decimal.
4. All the zeroes before a non-zero digit will be non-significant.

The exact number is a quantity which is obtained by definition or by counting. The measured number is a quantity which is measured through the measuring device. The measured quantities have some uncertainty and so they can have ambiguity.

Answer to Problem 27QAP

The correct calculation is $1.5\times {10}^{-22}$.

Explanation of Solution

Rules for significant figures:

1. For addition and subtraction, the calculation results should have same decimal places as found in the number with least significant digits in the problem.
2. For division and multiplication, the calculation results should have significant number as found in the number with least significant digits in the problem.

As per significant rule 2, the number with least significant digits will be used to decide the significant digits of result.

Firstly, looking at all the three numbers, the least significant digit is up to two (3.1×10-7).

Hence, the calculation result will be made up to correct significant digits as follows:

$x=\frac{(2.998\times {10}^8)(3.1\times {10}^{-7})}{6.022\times {10}^{23}}=\frac{9.3\times {10}^1}{6.022\times {10}^{23}}=1.5\times {10}^{-22}$

The result has two significant digits.

Hence, the calculation result will be $1.5\times {10}^{-22}$.