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Chapter 1: Problem 12

The volume of a solid cylinder is given by \(V=\pi r^{2} h,\) where \(r\) is the radius and \(h\) is the height. You measure the radius and height of a thin cylindrical wire and obtain the results \(r=0.036 \mathrm{~cm}\) and \(h=12.1 \mathrm{~cm}\). What do your measurements give for the volume of the wire in \(\mathrm{mm}^{3} ?\) Use the correct number of significant figures in your answer.

Short Answer

Expert verified
Using these steps, a volume in \(mm^3\) will be calculated, and this value, when rounded to three significant figures, is the answer.

Step by step solution

01

Insert Given Values into Volume Formula

Insert the given radius \(r=0.036\,\mathrm{cm}\) and height \(h=12.1\,\mathrm{cm}\) into the volume formula: \(V=\pi(0.036)^{2}(12.1).\)
02

Calculate the Volume in \(cm^3\)

By calculating the given formula, the volume \(V\) in \(cm^3\) should be found. Remember to square the radius before multiplying by \(\pi\) and the height.
03

Convert the Volume to \(mm^3\)

Convert the volume from \(cm^3\) to \(mm^3\) knowing that \(1\,cm^3 = 1000\,mm^3.\) Multiply the volume calculated in \(cm^3\) with 1000 to get the result in \(mm^3\).
04

Give the Answer Using Correct Number of Significant Figures

The provided values have three significant figures, so the final answer should be rounded to or expressed with three significant figures.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significant Figures
Understanding significant figures is crucial for reporting the precision of measurements in scientific calculations. Significant figures, or 'sig figs' as they're sometimes called, reflect the meaningful digits in a number that contribute to its precision. This includes all the digits from the first non-zero digit to the last non-zero digit or the last digit if it's estimated. For instance, in the radius measurement of 0.036 cm, all three digits are significant because they provide specific information about the precision of the measurement.

When performing mathematical operations like multiplication or division, the rule of thumb is to round the final result to the same number of significant figures as the least precise measurement. In our cylindrical wire example, both the radius and height (0.036 cm and 12.1 cm, respectively) are measured to three significant figures, therefore, the resulting volume should be reported with three significant figures. Precision in scientific reporting is critical, as overstating the accuracy of measurements can lead to misleading conclusions.
Unit Conversion
Unit conversion is a methodical process for converting the measurement of a quantity from one unit to another. In the realm of physics and engineering, it is common to encounter tasks requiring the conversion of volumes from cubic centimeters to cubic millimeters. The key to mastering unit conversion lies in understanding the relationship between different units of measurement. Specifically, for volume measurements:
  • 1 cubic centimeter (cm^3) is equal to 1000 cubic millimeters (mm^3).
To carry out the conversion, you multiply the value in cubic centimeters by 1000 to obtain the volume in cubic millimeters. This step is essential for expressing measurements in the preferred or required unit and ensuring consistency when comparing with other measurements. In educational settings, practicing unit conversion helps students develop a better grasp of scale and the significance of different measurement units in scientific contexts.
Cylindrical Volume Calculation
Calculating cylindrical volume is a routine task in mathematics and various real-world applications. The formula for finding the volume of a cylinder is represented as \( V = \pi r^2 h \), where \( V \) stands for volume, \( r \) denotes the radius of the circular base, and \( h \) symbolizes the height of the cylinder. The key steps to calculate the volume of a cylinder, such as a thin cylindrical wire, include:
  • Accurately measuring the radius and height of the cylinder.
  • Squaring the radius measurement.
  • Multiplying the squared radius by \( \pi \) (approximately 3.14159).
  • Multiplying this result by the height of the cylinder to get the volume.
When completing the calculations, it's essential to keep in mind the correct use of significant figures and unit conversion, ensuring the final volume is expressed accurately. It's a great exercise for students to familiarize themselves with geometric shapes and their properties, enhancing their spatial awareness and problem-solving skills.

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Most popular questions from this chapter

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