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 0.360094451 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 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?

evidence notebook explain how neglecting to account for significant figures could lead to major variations when using data from the analysis of a small sample of food to determine the amount of energy in a very large food sample.

Explanation:

Step1: Recall density - mass formula

The formula for density $
ho=\frac{m}{V}$, where $
ho$ is density, $m$ is mass and $V$ is volume. We can re - arrange it to $m =
ho V$.

Step2: Identify values and calculate mass for question 1

Given $V = 1062\ mL$ and $
ho=0.023\ g/mL$. Since multiplication is involved, and 0.023 has 2 significant figures while 1062 has 4 significant figures, the result should have 2 significant figures. $m=
ho V=0.023\ g/mL\times1062\ mL = 24.426\ g\approx24\ g$.

Step3: Calculate depth difference for question 2

For subtraction, $3.29\ m-1.0487\ m = 2.2413\ m$. The least precise measurement is 3.29 with 2 decimal places. So the result should be reported with 2 decimal places: $2.24\ m$.

Step4: Convert kilometers to meters for question 3

We know that $1\ km = 1000\ m$. For $0.704\ km$, using the conversion factor $0.704\ km\times1000\ m/km = 704\ m$. Since 1000 is an exact number, the result has 3 significant figures like 0.704.

Step5: Calculate cube volume for question 4

The volume of a cube $V = s^{3}$, where $s = 2.70\ cm$. $V=(2.70\ cm)^{3}=19.683\ cm^{3}$. Since 2.70 has 3 significant figures, the result should have 3 significant figures: $19.7\ cm^{3}$.

Answer:

1. $24\ g$
2. $2.24\ m$
3. $704\ m$
4. $19.7\ cm^{3}$