Chapter 1, Problem 1.76E

Interpretation Introduction

(a)

Interpretation:

The area and perimeter for each rectangle is to be calculated and the results are to be expressed in square centimeters and centimeters respectively. The correct number of significant figures is to be stated.

Concept introduction:

In a number, the digits which express the accurate measurement of the number are known as significant figures. It is counted starting from the first non-zero digit in the given number. Zeroes that lie between two non-zero digits are considered significant. Zeroes at the end of a number are considered non significant in the absence of a decimal point, whereas they are considered significant if there is a decimal point after it. Zeroes after a decimal point are considered significant.

Answer to Problem 1.76E

The areas of black, red, green and orange rectangles are $124.8{\text{cm}}^{\text{2}}$, $48.884{\text{cm}}^{\text{2}}$, $8.109{\text{cm}}^{\text{2}}$ and $8.9896{\text{cm}}^{\text{2}}$ respectively. The number of significant digits in the results is $4$, $5$, $4$ and $5$ respectively. The perimeters of black, red, green and orange rectangles are $44.8\text{cm}$, $45.24\text{cm}$, $11.46\text{cm}$ and $27.8\text{cm}$ respectively. The number of significant digits in the results is $3$, $4$, $4$ and $3$ respectively.

Explanation of Solution

Given

The length and width of the black rectangle are $l=12.00\text{cm}$ and $w=10.40\text{cm}$ respectively.

The length and width of the red rectangle are $l=20.20\text{cm}$ and $w=2.42\text{cm}$ respectively.

The length and width of the green rectangle are $l=3.18\text{cm}$ and $w=2.55\text{cm}$ respectively.

The length and width of the orange rectangle are $l=13.22\text{cm}$ and $w=0.68\text{cm}$ respectively.

The formula to calculate the area of a rectangle is given below.

$\text{Area}=l\times w$ …(1)

Where,

• $l$ is the length

• $w$ is the width

The area of black rectangle is calculated as shown below.

$\begin{array}{rll}\text{Area}& =& l\times w\\ & =& 12.00\times 10.40\\ & =& 124.8{\text{cm}}^{\text{2}}\end{array}$

The number of significant digits in the result is $4$.

The area of red rectangle is calculated as shown below.

$\begin{array}{rll}\text{Area}& =& l\times w\\ & =& 20.20\times 2.42\\ & =& 48.884{\text{cm}}^{\text{2}}\end{array}$

The number of significant digits in the result is $5$.

The area of green rectangle is calculated as shown below.

$\begin{array}{rll}\text{Area}& =& l\times w\\ & =& 3.18\times 2.55\\ & =& 8.109{\text{cm}}^{\text{2}}\end{array}$

The number of significant digits in the result is $4$.

The area of orange rectangle is calculated as shown below.

$\begin{array}{rll}\text{Area}& =& l\times w\\ & =& 13.22\times 0.68\\ & =& 8.9896{\text{cm}}^{\text{2}}\end{array}$

The number of significant digits in the result is $5$.

The formula to calculate the perimeter of a rectangle is given below.

$\begin{array}{rll}\text{Perimeter}& =& \text{Sum}\text{of}\text{all}\text{four}\text{sides}\\ & =& 2\left(l+w\right)\end{array}$ …(2)

Where,

• $l$ is the length

• $w$ is the width

The perimeter of black rectangle is calculated as shown below.

$\begin{array}{rll}\text{Perimeter}& =& 2\left(l+w\right)\\ & =& 2\left(12.00+10.40\right)\\ & =& 44.8\text{cm}\end{array}$

The number of significant digits in the result is $3$.

The perimeter of red rectangle is calculated as shown below.

$\begin{array}{rll}\text{Perimeter}& =& 2\left(l+w\right)\\ & =& 2\left(20.20+2.42\right)\\ & =& 45.24\text{cm}\end{array}$

The number of significant digits in the result is $4$.

The perimeter of green rectangle is calculated as shown below.

$\begin{array}{rll}\text{Perimeter}& =& 2\left(l+w\right)\\ & =& 2\left(3.18+2.55\right)\\ & =& 11.46\text{cm}\end{array}$

The number of significant digits in the result is $4$.

The perimeter of orange rectangle is calculated as shown below.

$\begin{array}{rll}\text{Perimeter}& =& 2\left(l+w\right)\\ & =& 2\left(13.22+0.68\right)\\ & =& 27.8\text{cm}\end{array}$

The number of significant digits in the result is $3$.

Conclusion

The areas of black, red, green and orange rectangles are $124.8{\text{cm}}^{\text{2}}$, $48.884{\text{cm}}^{\text{2}}$, $8.109{\text{cm}}^{\text{2}}$ and $8.9896{\text{cm}}^{\text{2}}$ respectively. The number of significant digits in the results is $4$, $5$, $4$ and $5$ respectively. The perimeters of black, red, green and orange rectangles are $44.8\text{cm}$, $45.24\text{cm}$, $11.46\text{cm}$ and $27.8\text{cm}$ respectively. The number of significant digits in the results is $3$, $4$, $4$ and $3$ respectively.

Interpretation Introduction

(b)

Interpretation:

The area and perimeter for each rectangle is to be calculated by changing all the measured values to meters and the results are to be expressed in square meters and meters respectively. The correct number of significant figures is to be stated.

Concept introduction:

In a number, the digits which express the accurate measurement of the number are known as significant figures. It is counted starting from the first non-zero digit in the given number. Zeroes that lie between two non-zero digits are considered significant. Zeroes at the end of a number are considered non significant in the absence of a decimal point, whereas they are considered significant if there is a decimal point after it. Zeroes after a decimal point are considered significant.

Answer to Problem 1.76E

The areas of black, red, green and orange rectangles are $0.01248{\text{m}}^{\text{2}}$, $0.0048884{\text{m}}^{\text{2}}$, $0.0008109{\text{m}}^{\text{2}}$ and $0.00089896{\text{m}}^{\text{2}}$ respectively. The number of significant digits in the results is $4$, $5$, $4$ and $5$ respectively. The perimeters of black, red, green and orange rectangles are $0.448\text{m}$, $0.4524\text{m}$, $0.1146\text{m}$ and $0.278\text{m}$ respectively. The number of significant digits in the results is $3$, $4$, $4$ and $3$ respectively.

Explanation of Solution

It is known that $\text{1}\text{cm}=\frac{\text{1}}{\text{100}}\text{m}$.

The conversion of units of the dimensions of black rectangle is shown below.

$\begin{array}{rll}l& =& 12.00\text{cm}\\ & =& 12.00\text{cm}\times \frac{\text{1}\text{m}}{\text{100}\text{cm}}& =& 0.12\text{m}\\ w& =& 10.40\text{cm}\\ & =& 10.40\text{cm}\times \frac{\text{1}\text{m}}{\text{100}\text{cm}}& =& 0.1040\text{m}\end{array}$

The conversion of units of the dimensions of red rectangle is shown below.

$\begin{array}{rll}l& =& 20.20\text{cm}\\ & =& 20.20\text{cm}\times \frac{\text{1}\text{m}}{\text{100}\text{cm}}& =& 0.2020\text{m}\\ w& =& 2.42\text{cm}\\ & =& 2.42\text{cm}\times \frac{\text{1}\text{m}}{\text{100}\text{cm}}& =& 0.0242\text{m}\end{array}$

The conversion of units of the dimensions of green rectangle is shown below.

$\begin{array}{rll}l& =& 3.18\text{cm}\\ & =& 3.18\text{cm}\times \frac{\text{1}\text{m}}{\text{100}\text{cm}}& =& 0.0318\text{m}\\ w& =& 2.55\text{cm}\\ & =& 2.55\text{cm}\times \frac{\text{1}\text{m}}{\text{100}\text{cm}}& =& 0.0255\text{m}\end{array}$

The conversion of units of the dimensions of orange rectangle is shown below.

$\begin{array}{rll}l& =& 13.22\text{cm}\\ & =& 13.22\text{cm}\times \frac{\text{1}\text{m}}{\text{100}\text{cm}}& =& 0.1322\text{m}\\ w& =& 0.68\text{cm}\\ & =& 0.68\text{cm}\times \frac{\text{1}\text{m}}{\text{100}\text{cm}}& =& 0.0068\text{m}\end{array}$

The formula to calculate the area of a rectangle is given below.

$\text{Area}=l\times w$ …(1)

Where,

• $l$ is the length

• $w$ is the width

The area of black rectangle is calculated as shown below.

$\begin{array}{rll}\text{Area}& =& l\times w\\ & =& 0.12\times 0.1040\\ & =& 0.01248{\text{m}}^{\text{2}}\end{array}$

The number of significant digits in the result is $4$.

The area of red rectangle is calculated as shown below.

$\begin{array}{rll}\text{Area}& =& l\times w\\ & =& 0.2020\times 0.0242\\ & =& 0.0048884{\text{m}}^{\text{2}}\end{array}$

The number of significant digits in the result is $5$.

The area of green rectangle is calculated as shown below.

$\begin{array}{rll}\text{Area}& =& l\times w\\ & =& 0.0318\times 0.0255\\ & =& 0.0008109{\text{m}}^{\text{2}}\end{array}$

The number of significant digits in the result is $4$.

The area of orange rectangle is calculated as shown below.

$\begin{array}{rll}\text{Area}& =& l\times w\\ & =& 0.1322\times 0.0068\\ & =& 0.00089896{\text{m}}^{\text{2}}\end{array}$

The number of significant digits in the result is $5$.

The formula to calculate the perimeter of a rectangle is given below.

$\begin{array}{c}\text{Perimeter}=\text{Sum}\text{of}\text{all}\text{four}\text{sides}\\ =2\left(l+w\right)\end{array}$ …(2)

Where,

• $l$ is the length

• $w$ is the width

The perimeter of black rectangle is calculated as shown below.

$\begin{array}{rll}\text{Perimeter}& =& 2\left(l+w\right)\\ & =& 2\left(0.1200+0.1040\right)\\ & =& 0.448\text{m}\end{array}$

The number of significant digits in the result is $3$.

The perimeter of red rectangle is calculated as shown below.

$\begin{array}{rll}\text{Perimeter}& =& 2\left(l+w\right)\\ & =& 2\left(0.2020+0.0242\right)\\ & =& 0.4524\text{m}\end{array}$

The number of significant digits in the result is $4$.

The perimeter of green rectangle is calculated as shown below.

$\begin{array}{rll}\text{Perimeter}& =& 2\left(l+w\right)\\ & =& 2\left(0.0318+0.0255\right)\\ & =& 0.1146\text{m}\end{array}$

The number of significant digits in the result is $4$.

The perimeter of orange rectangle is calculated as shown below.

$\begin{array}{rll}\text{Perimeter}& =& 2\left(l+w\right)\\ & =& 2\left(0.1322+0.0068\right)\\ & =& 0.278\text{m}\end{array}$

The number of significant digits in the result is $3$.

Conclusion

The areas of black, red, green and orange rectangles are $0.01248{\text{m}}^{\text{2}}$, $0.0048884{\text{m}}^{\text{2}}$, $0.0008109{\text{m}}^{\text{2}}$ and $0.00089896{\text{m}}^{\text{2}}$ respectively. The number of significant digits in the results is $4$, $5$, $4$ and $5$ respectively. The perimeters of black, red, green and orange rectangles are $0.448\text{m}$, $0.4524\text{m}$, $0.1146\text{m}$ and $0.278\text{m}$ respectively. The number of significant digits in the results is $3$, $4$, $4$ and $3$ respectively.

Interpretation Introduction

(c)

Interpretation:

Whether the number of significant figures gets changed on changing the units is to be stated.

Concept introduction:

In a number, the digits which express the accurate measurement of the number are known as significant figures. It is counted starting from the first non-zero digit in the given number. Zeroes that lie between two non-zero digits are considered significant. Zeroes at the end of a number are considered non significant in the absence of a decimal point, whereas they are considered significant if there is a decimal point after it. Zeroes after a decimal point are considered significant.

Answer to Problem 1.76E

The number of significant figures does not change when the units of area and perimeter are changed.

Explanation of Solution

The number of significant figures does not change when the units of area and perimeter are changed from square centimeters and centimeters to square meters and meters respectively. Exact numbers possess infinite number of significant digits and these digits are not uncertain. Therefore, the number of significant digits in the final answer is not affected and they remain the same even after conversion of units.

Conclusion

The number of significant figures is not affected when the units of area and perimeter are changed from square centimeters and centimeters to square meters and meters respectively.