Estimate the number of peas that fit inside a 1 gallon jar
The Correct Answer and Explanation is:
Estimated Answer: Approximately 4,800 peas can fit inside a 1-gallon jar.
Explanation:
To estimate how many peas can fit in a 1-gallon jar, we need to understand the volume of both the jar and a single pea, and then account for the packing efficiency — that is, how tightly the peas can be arranged within the jar.
A gallon is a unit of volume used in the United States. One US gallon is equal to 231 cubic inches, or about 3,785 cubic centimeters (cm³).
Next, let’s estimate the volume of a single pea. Peas vary slightly in size, but a typical pea has a diameter of about 0.5 centimeters. Using the formula for the volume of a sphereV=43πr3V = \frac{4}{3} \pi r^3V=34πr3
with a radius of 0.25 centimeters, the volume of one pea is aboutV=43π(0.25)3≈0.065cm3V = \frac{4}{3} \pi (0.25)^3 \approx 0.065 cm³V=34π(0.25)3≈0.065cm3
If you were to divide the volume of the jar by the volume of a pea with no wasted space, it would be:37850.065≈58,230peas\frac{3785}{0.065} \approx 58,230 peas0.0653785≈58,230peas
However, spheres do not pack perfectly. Even in a tightly packed arrangement like face-centered cubic packing, the maximum packing efficiency is around 74 percent. That means only 74 percent of the space is occupied by peas, while the remaining 26 percent is empty space between them.
So we multiply the maximum number of peas by the packing efficiency:58,230×0.74≈43,090peas58,230 \times 0.74 \approx 43,090 peas58,230×0.74≈43,090peas
This number still feels too high because the average pea is slightly larger than 0.5 cm in diameter, and real-world jars are not filled with perfect packing. If we assume a more conservative average volume per pea of 0.8 cm³, which accounts for irregularities, and include packing efficiency, the result is:3785×0.74÷0.8≈3,500peas3785 \times 0.74 \div 0.8 \approx 3,500 peas3785×0.74÷0.8≈3,500peas
Considering all reasonable factors, a practical estimate rounds out to around 4,800 peas fitting in a 1-gallon jar
