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The thickness of the numbered pages of a textbook is measured to be \(3.75 \mathrm{~cm}\). (a) If the last page of the book is numbered 860 , what is the average thickness of a page? (b) Repeat the calculation by using order-of- magnitude calculations.

Short Answer

Expert verified
The average thickness is approximately 0.00872 cm.

Step by step solution

01

Calculate Total Number of Pages

There are pages on both sides of each numbered sheet. Since 860 is the number on the last page, we have half as many sheets. Thus, the number of sheets is \( \frac{860}{2} = 430 \).
02

Find Average Thickness per Page

Divide the total thickness by the number of sheets to find the average thickness: \( \frac{3.75 \mathrm{~cm}}{430} = 0.00872 \mathrm{~cm} \).
03

Order-of-Magnitude Approximation of Total Sheets

For an order of magnitude estimate, approximate the number of sheets. 860 pages implies around 1000 pages, which gives roughly 500 sheets (as 1000/2 = 500).
04

Order-of-Magnitude Average Thickness

Perform the order-of-magnitude division: \( \frac{3.75 \mathrm{~cm}}{500} = 0.0075 \mathrm{~cm} \).
05

Comparison of Exact and Order-of-Magnitude Results

The exact thickness per page is approximately 0.00872 cm, and the order-of-magnitude thickness is 0.0075 cm, showing a rough agreement within the same order of magnitude.

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Key Concepts

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

Order of Magnitude
When solving physics problems, it is essential to grasp the concept of "Order of Magnitude". This method provides an estimation of quantity by rounding to the nearest power of ten. It gives a rough idea or approximation rather than an exact number. Order of magnitude calculations are particularly helpful when one needs to make quick estimates without requiring precise data.
For instance, in our problem where we estimated the number of pages to be around 1000, even though the actual number was 860, it allowed us to simplify and work more comfortably with numbers. The idea is to round 860, the number on the last page, up to 1000, simplifying the scenario into easily understandable terms.
This method doesn’t give an exact value but helps us to stay within a reasonable range. It is common in physics to provide a sense of scale or magnitude for estimates, and it ensures that calculations remain feasible when precise measurements aren't available or necessary.
Average Thickness Calculation
To find the average thickness of a page, one needs to use precise numbers for an exact calculation. It involves dividing the total measured thickness by the number of sheets. In our exercise, the total thickness of the pages was given as 3.75 cm, and the number of sheets was deduced from the total number of pages.
Because each sheet of paper contributes two pages (one on each side), 860 pages make up 430 sheets. Therefore, calculating the average thickness per sheet involves dividing 3.75 cm by 430 sheets, resulting in approximately 0.00872 cm per sheet.
This precise calculation is more accurate than the order-of-magnitude approach, as it considers the actual measurements rather than approximations. Consequently, it is the preferred method when measurements must be accurate and reliable, ensuring precise, dependable results.
Measurement Techniques
Measurement techniques are fundamental in physics to ensure that calculations, like those for average thickness, are as accurate as possible. Several key techniques can enhance measurement reliability:
  • Use of precision instruments such as calipers and micrometers when measuring thicknesses.
  • Ensuring multiple measurements are taken and averaged to mitigate any anomalies or errors.
  • Understanding the measurement unit scale, which should be appropriately chosen for the size of the object being measured.

These methods can be crucial when applying concepts like those in our exercise. For example, when measuring the thickness of a textbook, precision tools should capture the depth accurately. By improving measurement practices, like minimizing human error and using suitable equipment, the resulting data becomes more trustworthy.
Always align the measurement technique with the importance of the problem at hand. Simple tools and estimates, such as those used in order-of-magnitude evaluations, are fine for rough approximations, while detailed tasks benefit from refined measurement techniques.

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Most popular questions from this chapter

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