Let's break down the problem: **Question:** A specimen is observed under a microscope with a magnification of 300. Over a 3 in² region, the specimen has about 60 grains. Calculate the ASTM grain size number for the material. --- ### ASTM Grain Size Number Formula The ASTM grain size number (n) is given by: \[ n = 1 + \log_2 N \] Where \(N\) is the number of grains per square inch at 100× magnification. Given: - Area = 3 in² - Number of grains in that area = 60 - Magnification = 300× First, find the number of grains per square inch at 300×: \[ N_{300} = \frac{60}{3} = 20 \text{ grains/in}^2 \text{ at 300×} \] Now convert this to the equivalent number at 100×: \[ N_{100} = N_{300} \left(\frac{300}{100}\right)^2 = 20 \times 9 = 180 \text{ grains/in}^2 \text{ at 100×} \] Now, plug into the ASTM formula: \[ n = 1 + \log_2(180) \] \[ \log_2(180) = \frac{\log_{10}(180)}{\log_{10}(2)} \approx \frac{2.2553}{0.3010} \approx 7.495 \] \[ n = 1 + 7.495 = 8.495 \] --- ### Closest Answer From the options, **7.5** is the closest and most reasonable match. --- **Final Answer:** **7.5** Let me know if you need any clarifications or further explanation!