(a) How many significant figures should be reported for the...

(a) How many significant figures should be reported for the volume of the metal bar shown here? (b) If the mass of the bar is $104.72 \mathrm{~g}$, how many significant figures should be reported when its density is determined using the calculated volume? [Section 1.6]

Key Concepts

Propagation of Significant Figures in Calculations

When combining measurements through mathematical operations, the result cannot be more precise than the least precise measurement. This propagation rule means that in operations like multiplication or division, the result should be reported with the same number of significant figures as the measurement with the fewest significant figures.

Density Calculations from Measured Quantities

Density is calculated by dividing mass by volume. In such derived calculations, the precision of the density is limited by the measurement (mass or volume) with the fewest significant figures, so it is essential to consider significant figures when finalizing the result.

Significant Figures

Significant figures are the digits in a measurement that are known with certainty plus one final uncertain digit. They provide a way to communicate the precision of a measurement and ensure that calculations do not imply more precision than what the measurements support.

Measurement Precision and Instrument Limits

The precision of a measurement is dictated by the limitations of the measuring instrument, which in turn determines the number of significant figures reported. Understanding the instrument’s resolution is crucial for reporting values such as lengths, masses, or volumes accurately.

Transcript

00:01 Talking about significant figures and how we'll be able to apply significant figures, the concept of it, when we're performing calculations.

00:09 So significant figures here have written a definition.

00:12 They're each of the digits and a number that are used to express the accuracy of a calculation or measurement.

00:17 So they're really crucial when you are dividing, multiplying, adding, subtracting.

00:23 So that will really affect what you're going to get as your final answer.

00:27 So here we have this little diagram here where we're trying to find the volume of this particular metal bar per se.

00:38 And so what we want to do, again, the formula for volume is length times width times height.

00:43 So if we want to go ahead and take a look at the volume here, we want to calculate it, let's go ahead and just multiply each of the numbers together.

00:51 So we can have 5 .30 times 2 .5 times 1 .25.

01:04 So when we go ahead and plug that all into the calculator, you'll see that we're going to get about 16 .56 centimeters cubed.

01:13 Let me go ahead and put an error here to indicate that is cute.

01:16 Now, that's not going to be our final answer if we are considering significant figures.

01:20 So between all the three numbers that we have here, our 2 .5 has the least number of significant figures.

01:28 It has two.

01:30 In comparison to the rest, it has three.

01:34 So because our 2 .5 has the least number of significant figures, which has two, we're going to have to apply that to our final answer.

01:41 So we have to go by the product or one of our numbers that has the least number of significant digits.

01:47 So if we have two, then therefore our final answer has to have two significant figures...

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