Chapter 2, Problem 92A

(a)

Interpretation Introduction

Interpretation:

The answer of given problem needs to be rounded off to the correct significant figures.

$\left(7.31\times {10}^4\right)+\left(3.23\times {10}^3\right)$

Concept introduction:

In order to determine the number of significant figures, the following rules should be applied,

1. The non-zero digits are always significant.

2. In a number, the zeros to the left of the first non-zero digit are not significant.

3. Zeros between non zero digits are significant

4. Zeros to the right of the decimal place are significant

5. If a number ends in zero that are not to the right of a decimal,  the zeros may or may not be significant.

To round off a number, look at the digit up to which number needs to be rounded off. If the digit right to it is less than 5 simply add 1 to the previous digit and remove all the digits after it from the number. If the digit right to the digit up to which rounding needs to be done is less than 5, just remove all the digits after it from the number.

Now, if the digit is equal to 5 then 1 is added to the previous digit if it is an odd number. If the previous digit is an even number, simply remove all the digits after it from the number.

Answer to Problem 92A

The final answer of the given problem is correct to $\text{7}\text{.63}\times {10}^4$.

Explanation of Solution

When adding the numbers in scientific notation, the exponent values must be same. Simply, add both numbers and keeps the exponent value same.

$\text{3}\text{.23}\times {10}^3=0.\text{323}\times {10}^4$

$\begin{array}{c}\left(7.31\times {10}^4\right)+\left(0.323\times {10}^4\right)=\left(7.31+0.323\right)\times {10}^4\\ \text{=}\text{7}\text{.633}\times {10}^4\\ =\text{7}\text{.63}\times {10}^4\end{array}$

Number of significant figures of the final answer can be determined by the original value having least digits after the decimal point. Therefore, the final answer of the given solution will contain digit that goes to the hundredths place. Hence, the correct value is $\text{7}\text{.63}\times {10}^4$.

(b)

Interpretation Introduction

Interpretation:

The answer of given problem needs to be rounded off to the correct significant figures.

$\left(8.54\times {10}^{-3}\right)-\left(3.41\times {10}^{-4}\right)$

Concept introduction:

In order to determine the number of significant figures, the following rules should be applied,

1. The non-zero digits are always significant.

2. In a number, the zeros to the left of the first non-zero digit are not significant.

3. Zeros between non zero digits are significant

4. Zeros to the right of the decimal place are significant

5. If a number ends in zero that are not to the right of a decimal,  the zeros may or may not be significant.

To round off a number, look at the digit up to which number needs to be rounded off. If the digit right to it is less than 5 simply add 1 to the previous digit and remove all the digits after it from the number. If the digit right to the digit up to which rounding needs to be done is less than 5, just remove all the digits after it from the number.

Now, if the digit is equal to 5 then 1 is added to the previous digit if it is an odd number. If the previous digit is an even number, simply remove all the digits after it from the number.

Answer to Problem 92A

The final answer of the given problem is correct to $\text{8}\text{.20}\times {10}^{-3}$.

Explanation of Solution

When subtracting the numbers in scientific notation, the exponent values must be same. Simply, add both numbers and keeps the exponent value same.

$3.41\times {10}^{-4}=0.341\times {10}^{-3}$

$\begin{array}{c}\left(8.54\times {10}^{-3}\right)-\left(0.341\times {10}^{-3}\right)=\left(8.54-0.341\right)\times {10}^{-3}\\ \text{=}8.199\times {10}^{-3}\\ =\text{8}\text{.20}\times {10}^{-3}\end{array}$

Number of significant figures of the final answer can be determined by the original value having least digits after the decimal point. Therefore, the final answer of the given solution will contain digit that goes to the hundredths place. Hence, the correct value is $\text{8}\text{.20}\times {10}^{-3}$.

(c)

Interpretation Introduction

Interpretation:

The answer of given problem needs to be rounded off to the correct significant figures.

$4.35\text{dm}\times 2.34\text{dm}\times 7.35\text{dm}$

Concept introduction:

In order to determine the number of significant figures, the following rules should be applied,

1. The non-zero digits are always significant.

2. In a number, the zeros to the left of the first non-zero digit are not significant.

3. Zeros between non zero digits are significant

4. Zeros to the right of the decimal place are significant

5. If a number ends in zero that are not to the right of a decimal,  the zeros may or may not be significant.

To round off a number, look at the digit up to which number needs to be rounded off. If the digit right to it is less than 5 simply add 1 to the previous digit and remove all the digits after it from the number. If the digit right to the digit up to which rounding needs to be done is less than 5, just remove all the digits after it from the number.

Now, if the digit is equal to 5 then 1 is added to the previous digit if it is an odd number. If the previous digit is an even number, simply remove all the digits after it from the number.

Answer to Problem 92A

The final answer of the given problem is correct to $74.8{\text{dm}}^{\text{3}}$.

Explanation of Solution

All the three value in the given problem have the same number of significant figures. Therefore, the solution will contain three significant figures. Hence the correct value is $74.8{\text{dm}}^{\text{3}}$.

$\begin{array}{c}4.35\text{dm}\times 2.34\text{dm}\times 7.35\text{dm}=74.81565{\text{dm}}^{\text{3}}\\ =74.8{\text{dm}}^{\text{3}}\end{array}$

(d)

Interpretation Introduction

Interpretation:

The answer of given problem needs to be rounded off to the correct significant figures.

$4.78\text{cm}+3.218\text{cm}+5.82\text{cm}$

Concept introduction:

In order to determine the number of significant figures, the following rules should be applied,

1. The non-zero digits are always significant.

2. In a number, the zeros to the left of the first non-zero digit are not significant.

3. Zeros between non zero digits are significant

4. Zeros to the right of the decimal place are significant

5. If a number ends in zero that are not to the right of a decimal,  the zeros may or may not be significant.

To round off a number, look at the digit up to which number needs to be rounded off. If the digit right to it is less than 5 simply add 1 to the previous digit and remove all the digits after it from the number. If the digit right to the digit up to which rounding needs to be done is less than 5, just remove all the digits after it from the number.

Now, if the digit is equal to 5 then 1 is added to the previous digit if it is an odd number. If the previous digit is an even number, simply remove all the digits after it from the number.

Answer to Problem 92A

The final answer of the given problem is correct to $13.8\text{cm}$.

Explanation of Solution

Number of significant figures of the final answer can be determined by the original value having least digits after the decimal point. Therefore, the final answer of the given solution will contain three significant figures. Hence, the correct value is $13.8\text{cm}$.

$\begin{array}{c}4.78\text{cm}+3.218\text{cm}+5.82\text{cm}=13.818\text{cm}\\ =13.8\text{cm}\end{array}$

(e)

Interpretation Introduction

Interpretation:

The answer of given problem needs to be rounded off to the correct significant figures.

$38,736\text{km}÷4784\text{km}$

Concept introduction:

In order to determine the number of significant figures, the following rules should be applied,

1. The non-zero digits are always significant.

2. In a number, the zeros to the left of the first non-zero digit are not significant.

3. Zeros between non zero digits are significant

4. Zeros to the right of the decimal place are significant

5. If a number ends in zero that are not to the right of a decimal,  the zeros may or may not be significant.

To round off a number, look at the digit up to which number needs to be rounded off. If the digit right to it is less than 5 simply add 1 to the previous digit and remove all the digits after it from the number. If the digit right to the digit up to which rounding needs to be done is less than 5, just remove all the digits after it from the number.

Now, if the digit is equal to 5 then 1 is added to the previous digit if it is an odd number. If the previous digit is an even number, simply remove all the digits after it from the number.

Answer to Problem 92A

The final answer of the given problem is correct to $8.097$.

Explanation of Solution

When dividing the numbers, answer must have the same number of the significant figures as the measurement with the lowest number of significant figures. The lowest number of the significant figure from the given two values is 4. Hence, the solution must contain 4 significant figures.

$\begin{array}{c}38,736\text{km}÷4784\text{km}\text{=}\frac{38,736\text{km}}{4784\text{km}}\\ =8.0969899\\ =8.097\end{array}$

Hence, the correct value is $8.097$.