Final answer:
To find the volume of the sandcastle, calculate the volume of a single sand grain and multiply it by Avogadro's number. Then convert the volume to km^3. Next, divide the volume by the cross-sectional area of the beach to find the length it would cover.
Step-by-step explanation:
To find the volume of the sandcastle, we need to calculate the volume of a single sand grain and then multiply it by Avogadro's number. Each grain is a cube with sides measuring 1.0 mm. The volume of a cube is found by cubing the length of its side, so the volume of a single grain is (1.0 mm)^3 = 0.001 mm^3. Avogadro's number is approximately 6.022 x 10^23, so the volume of Avogadro's number of sand grains is (0.001 mm^3) x (6.022 x 10^23) = 6.022 x 10^20 mm^3.
To convert this volume to km^3, we need to divide by the appropriate conversion factor. There are 1,000,000 mm^3 in 1 cm^3, and 1,000,000 cm^3 in 1 m^3. Finally, there are 1,000,000,000 m^3 in 1 km^3. So, to convert from mm^3 to km^3, we divide by (1,000,000 mm^3/cm^3) x (1,000,000 cm^3/m^3) x (1,000,000,000 m^3/km^3) = 1 x 10^18. Dividing 6.022 x 10^20 mm^3 by 1 x 10^18 gives us 6.022 x 10^2 km^3.
Now, let's calculate the length of the beach that this volume of sand would cover. The beach has an average width of 100 m and depth of 10.0 m. To find the length, we divide the volume (6.022 x 10^2 km^3) by the cross-sectional area (100 m x 10.0 m = 1000 m^2). Dividing 6.022 x 10^2 km^3 by 1000 m^2 gives us 6.022 x 10^ -1 kilometers, which is equal to 0.6 kilometers.