Question

rules for using significant figures in mathematical operations:

• multiplication and division: the number of significant figures in the result is the same as the number in the least precise measurement used in the calculation. for example, 3.05 divided by 8.470 will produce a result of 0.360 094 451 on a calculator. the result, however, should only have three significant figures, so the final value should be correctly reported as 0.360.
• addition and subtraction: the result has the same number of decimal places as the least precise measurement used in the calculation. for example, 25.1 added to 2.03 will produce a result of 27.13 on a calculator. however, the value 25.1 has only one decimal place, so the value in the hundredths place is completely unknown. the final answer should therefore be reported with one decimal place: 27.1.
• exact numbers, such as the conversion factor between meters and centimeters, do not limit significant figures, as they are considered to have no uncertainty. for example:

4.608 m × \\( \frac { 100 cm } { 1 m } \\) = 460.8 cm
solve report the result of each calculation using the correct number of significant figures.

1. a scientist measures 1062 ml of a substance with a density of 0.023 g/ml. what is the mass of the substance?
2. an engineer measures the initial depth of liquid in a reactor vessel as 3.29 m and the final depth as 1.0487 m. what is the difference in depth?
3. the distance between two points on a map is 0.704 kilometers. what is this distance in meters?
4. a student measures the the sides of a cube with a ruler and finds that the length of each side is 2.70 cm. what is the volume of the cube in centimeters cubed?

Explanation:

Step1: Recall the formula for mass

The formula for mass \(m\) is \(m =
ho V\), where \(
ho\) is density and \(V\) is volume.

Step2: Substitute the given values

Given \(V = 1062\space mL\) and \(
ho=0.023\space g/mL\), then \(m=(0.023\space g/mL)\times(1062\space mL)\)

$$m = 0.023\times1062=24.426\space g$$

Step3: Apply the rule for multiplication/division (significant figures)

The number \(0.023\) has two significant figures and \(1062\) has four significant figures. According to the rule for multiplication/division (the result has the same number of significant figures as the least - precise measurement), we round \(24.426\) to two significant figures.

Step4: Recall the formula for subtraction

For the second problem, the formula for the difference \(\Delta h=h_1 - h_2\), where \(h_1 = 3.29\space m\) and \(h_2 = 1.0487\space m\)

$$h_1 - h_2=3.29 - 1.0487=2.2413\space m$$

Step5: Apply the rule for subtraction (significant figures)

The number \(3.29\) has two decimal places and \(1.0487\) has four decimal places. According to the rule for addition/subtraction (the result has the same number of decimal places as the least - precise measurement), we round \(2.2413\) to two decimal places.

Step6: Use the conversion factor for kilometers to meters

We know that \(1\space km = 1000\space m\). For the third problem, if \(d = 0.704\space km\), then \(d=(0.704\space km)\times(1000\space m/km)\)

$$d = 704\space m$$

The conversion factor \(1000\) (exact number) does not limit significant figures. The number \(0.704\) has three significant figures, and \(704\) also has three significant figures.

Step7: Recall the formula for the volume of a cube

The formula for the volume of a cube \(V = s^3\), where \(s = 2.70\space cm\)

$$V=(2.70)^3=2.70\times2.70\times2.70 = 19.683\space cm^3$$

Step8: Apply the rule for multiplication (significant figures)

The number \(2.70\) has three significant figures. According to the rule for multiplication/division (the result has the same number of significant figures as the least - precise measurement), we round \(19.683\) to three significant figures.

Answer:

1. \(24\space g\)
2. \(2.24\space m\)
3. \(704\space m\)
4. \(19.7\space cm^3\)