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Chapter 4: Problem 35

Determine the ASTM grain size number if 25 grains per square inch are measured at a magnification of \(600 \times\).

Short Answer

Expert verified
Answer: The approximate ASTM grain size number is 0.

Step by step solution

01

Convert the number of grains per square inch to a magnification of 100x.

Since we are given the number of grains per square inch at a magnification of 600x, we will need to convert it to the value at a magnification of 100x. Let's denote the number of grains per square inch at 600x magnification as \(N_{600}\) and the number of grains per square inch at 100x magnification as \(N_{100}\). The conversion can be done using the following formula: \(N_{100} = N_{600} \times (\frac{100}{600})^2\) Now plug in the given values to find \(N_{100}\): \(N_{100} = 25 \times (\frac{100}{600})^2\)
02

Calculate \(N_{100}\).

Calculate the value of \(N_{100}\) using the formula derived in Step 1: \(N_{100} = 25 \times (\frac{1}{6})^2 = 25 \times \frac{1}{36} = \frac{25}{36}\)
03

Solve for the ASTM grain size number (n).

Using the formula \(N = 2^{(n-1)}\), plug in the value of \(N_{100}\) and solve for n: \(\frac{25}{36} = 2^{(n-1)}\) To solve for n, first take the logarithm base 2 of both sides of the equation: \(log_2(\frac{25}{36}) = log_2(2^{(n-1)})\) Now use the fact that \(log_a(a^x) = x\): \(log_2(\frac{25}{36}) = n-1\) Finally, add 1 to both sides of the equation to find n: \(n = log_2(\frac{25}{36}) + 1\)
04

Calculate the ASTM grain size number (n).

Using the formula derived in Step 3, calculate the value of n: \(n = log_2(\frac{25}{36}) + 1 \approx -0.85 + 1 \approx 0.15\) Since the ASTM grain size number must be an integer, round the result to the nearest whole number: \(n \approx 0\) Thus, the ASTM grain size number is approximately 0. Keep in mind that this result may not be practically meaningful, so it is best to consult with a materials expert or refer to reference materials while working with the ASTM grain size number in practice.

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