About how much does a grain of sand weigh? How many grains of sand are there in a typical playground sandbox? The volume of a grain of sand is about 6.2×10^-5 mm^3. If a cylindrical bucket has a radius of 100 mm and a height of 200 mm, then the volume of the bucket is 3.14×10^7 mm^3. This means that there are about 5.06×10^11 grains of sand in the bucket. Assuming that the bucket weighs about 21 pounds, the weight of a grain of sand is about 4.15×10^-5 pounds. If there are about 100 pounds of sand in a typical playground sandbox, then there are about 2.41×10^6 grains of sand in a typical playground sandbox.
The Correct Answer and Explanation is:
To find the weight of a grain of sand and estimate how many grains of sand are in a typical playground sandbox, we need to use the provided information and some basic mathematical calculations.
1. Weight of a Grain of Sand
We are given that:
- A cylindrical bucket has a volume of 3.14×107 mm33.14 \times 10^7 \, \text{mm}^33.14×107mm3.
- The weight of the entire bucket of sand is 21 pounds.
- The volume of a grain of sand is 6.2×10−5 mm36.2 \times 10^{-5} \, \text{mm}^36.2×10−5mm3.
- There are approximately 5.06×10115.06 \times 10^{11}5.06×1011 grains of sand in the bucket.
Step 1: Find the weight of a grain of sand
The weight of a grain of sand can be calculated by dividing the total weight of the sand in the bucket by the number of grains:Weight of one grain of sand=Weight of sand in bucketNumber of grains in bucket=21 pounds5.06×1011≈4.15×10−5 pounds.\text{Weight of one grain of sand} = \frac{\text{Weight of sand in bucket}}{\text{Number of grains in bucket}} = \frac{21 \, \text{pounds}}{5.06 \times 10^{11}} \approx 4.15 \times 10^{-5} \, \text{pounds}.Weight of one grain of sand=Number of grains in bucketWeight of sand in bucket=5.06×101121pounds≈4.15×10−5pounds.
So, a single grain of sand weighs about 4.15×10−54.15 \times 10^{-5}4.15×10−5 pounds.
2. Grains of Sand in a Playground Sandbox
Next, we’re told that a typical playground sandbox contains about 100 pounds of sand. To estimate the number of grains of sand in the sandbox, we can divide the total weight of the sand by the weight of a single grain of sand:Number of grains in sandbox=Weight of sand in sandboxWeight of one grain of sand=100 pounds4.15×10−5 pounds≈2.41×106 grains.\text{Number of grains in sandbox} = \frac{\text{Weight of sand in sandbox}}{\text{Weight of one grain of sand}} = \frac{100 \, \text{pounds}}{4.15 \times 10^{-5} \, \text{pounds}} \approx 2.41 \times 10^6 \, \text{grains}.Number of grains in sandbox=Weight of one grain of sandWeight of sand in sandbox=4.15×10−5pounds100pounds≈2.41×106grains.
Thus, there are about 2.41 million grains of sand in a typical playground sandbox.
Conclusion
The weight of a single grain of sand is roughly 4.15×10−54.15 \times 10^{-5}4.15×10−5 pounds, and there are approximately 2.41 million grains of sand in a typical playground sandbox that weighs about 100 pounds.
