Problem 93 - Chemistry Matter and Change

Question

The accepted length of a steel pipe is 5.5 \(\mathrm{m} .\) Calculate the percent error for each of these measurements. a. 5.2 \(\mathrm{m} \quad\) b. 5.5 \(\mathrm{m}\) \(\quad\) c. 5.7 \(\mathrm{m} \quad\) d. 5.1 \(\mathrm{m}\)

Step-by-Step Solution

Answer

Percent errors: a. 5.45%, b. 0%, c. 3.64%, d. 7.27%

Step 1: Understand the Percent Error Formula

The formula to calculate percent error is given by:\[ \text{Percent Error} = \left( \frac{ |\text{Measured Value} - \text{Accepted Value}| }{ \text{Accepted Value} } \right) \times 100 \% \]Here, the accepted length of the steel pipe is 5.5 m.

Step 2: Calculate Percent Error for Measurement 5.2 m

Use the percent error formula:\[\text{Percent Error} = \left( \frac{ |5.2 - 5.5| }{ 5.5 } \right) \times 100 = \left( \frac{ 0.3 }{ 5.5 } \right) \times 100 \approx 5.45\% \]

Step 3: Calculate Percent Error for Measurement 5.5 m

Use the percent error formula:\[\text{Percent Error} = \left( \frac{ |5.5 - 5.5| }{ 5.5 } \right) \times 100 = 0\%\]The measured value is the same as the accepted value, so the percent error is 0%.

Step 4: Calculate Percent Error for Measurement 5.7 m

Use the percent error formula:\[\text{Percent Error} = \left( \frac{ |5.7 - 5.5| }{ 5.5 } \right) \times 100 = \left( \frac{ 0.2 }{ 5.5 } \right) \times 100 \approx 3.64\% \]

Step 5: Calculate Percent Error for Measurement 5.1 m

Use the percent error formula:\[\text{Percent Error} = \left( \frac{ |5.1 - 5.5| }{ 5.5 } \right) \times 100 = \left( \frac{ 0.4 }{ 5.5 } \right) \times 100 \approx 7.27\% \]

Key Concepts

Measurement Accuracy

Measurement accuracy is all about how close a measurement is to an accepted or true value. For example, if the accepted length of a steel pipe is 5.5 meters, a measurement of 5.5 meters is considered to be very accurate. Accuracy is crucial in the real world, especially in construction and manufacturing, where even small discrepancies can have significant impacts.

Consider the following aspects when thinking about measurement accuracy:

• True Value: Known or accepted standard value often derived from careful references.
• Measured Value: The value obtained from measurements using instruments or tools.
• Comparison: Accuracy involves comparing the measured value to the true value.

Achieving higher accuracy might involve using more precise instruments or ensuring that the measuring conditions (like temperature and alignment) minimize errors.

Steel Pipe Measurements

Measuring lengths of steel pipes accurately is essential in fields like construction and plumbing. Steel pipes must be measured precisely to ensure they fit into a given space or meet the needs of a project. Inaccuracies can lead to poor fitting, wasted materials, and potential safety hazards. Here's what you need to keep in mind when measuring steel pipes:

• Measure Consistently: Use the same measuring tools and techniques for consistency.
• Check the Units: Always verify the units you are measuring in, whether meters or feet.
• Account for Curvature: Ensure the measuring tape or device follows the pipe's length without bending away.

A well-measured pipe contributes directly to the project's success, minimizing errors and ensuring a seamless operation.

Percent Error Formula

Understanding the percent error formula is like having an essential tool in your math toolkit. Percent error helps compare your measured value to a known value, showing how close you are to the "true" value as a percentage. This is especially useful with steel pipe measurements, where understanding deviations can guide quality control.

The percent error formula looks like this:\[\text{Percent Error} = \left( \frac{ |\text{Measured Value} - \text{Accepted Value}| }{ \text{Accepted Value} } \right) \times 100 \%\]A few things to note about calculating percent error:

• The formula uses the absolute value to ensure any difference is positive.
• Multiplying by 100 converts the result into a percentage.
• Zero percent error indicates perfect accuracy.

This formula helps assess quality and precision, allowing you to refine methods or tools for more accurate results next time.