Question

The height of a picture frame is known to three significant figures, and the width is known to two significant figures. How many significant figures are there in the area of the picture frame?

Short answer

The area has two significant figures.

Step-by-step solution

Understanding Significant Figures

Significant figures are the meaningful digits in a number that contribute to its precision. When multiplying or dividing, the number of significant figures in the result is determined by the number used in the calculation with the fewest significant figures.

Analyze Given Data

The height of the picture frame is known to three significant figures and the width to two significant figures. This means that our calculation of the area will be limited to the smaller of these two, which is two significant figures.

Determine the Significant Figures for Area

Since the width has only two significant figures, when calculating the area of the picture frame by multiplying the height and width, the result must also be expressed with two significant figures to maintain the least precise measurement.

Key concepts

Multiplication Precision

When you multiply numbers, the precision of your product depends on the number of significant figures in the numbers you used. Let’s imagine you’re multiplying the height and width of a picture frame to find its area. Each of these dimensions has its own level of precision because they are measured values. Significant figures matter because they indicate how certain we are about measurements. For example, a height with three significant figures shows more precision than one with only two. When multiplying, you don’t take the total number of significant figures from both numbers; instead, the product will have the same number of significant figures as the factor with the fewest significant figures. This means for our picture frame area calculation, whatever number had the least precision (fewest significant figures), will limit the precision of the final area result. Here are some key points to remember:

• The product in a multiplication calculation should not have more significant figures than the least precise measurement.
• When precision differs, always round the result to match the least precise factor for consistency and accuracy.

Using these rules ensures your final result reflects the inherent uncertainty in the initial measurements.

Measurement Accuracy

Measurement accuracy is all about how close our measured values are to the true value or how many digits we trust in our numbers. In a perfect world, every measurement would be perfectly exact, but in reality, they are often approximations. In contexts such as calculating the area of a picture frame, the accuracy of each dimension (height and width) is vital. If, for instance, one measurement is accurate to two significant figures and another is to three, the least accurate measurement dictates the overall result's accuracy. This is because less accurate measurements introduce uncertainty into your final calculation. When measuring:

• Base your calculations on the measure known with least accuracy to ensure integrity and reliability.
• Always verify the measurement tools and how consistent they are to improve measurement accuracy.

Using the lowest common accuracy keeps your calculations safe from overstated precision, giving your results practical meaning and avoiding misrepresentation.

Calculation Limitations

Calculations have inherent limitations, especially when reliant on manual measurements. These limitations are tied to the potential errors in measurement and the precision of the numbers involved. In any calculation like determining the area of a picture frame, certain limitations come into play:

• Precision Limits: As seen, the results are only as precise as the least precise measurement.
• Rounding Errors: Typically, rounding can lead to small deviations in expected outcomes, particularly if significant figures are not appropriately maintained.
• Instrumental Precision: The tools you use can introduce limitations. For example, a ruler marked only in centimeters lacks the precision of one marked in millimeters.

Understanding these limitations helps in producing more realistic and accurate results. Always acknowledge the limits of precision and measurement accuracy before finalizing your answers, as overlooking them can lead to errors and misconceptions in interpreting data.