To estimate how many average-sized South American bodies would be needed to form a pile 19 feet long and 21 feet wide, with the bodies stacked in the middle and cascading down to contribute to the pile's width, we need to make some assumptions and perform rough calculations. The bodies are assumed to be stacked in a way that forms a roughly conical or pyramidal pile, with the height determined by the stacking arrangement.
Assumptions:
- Average Body Size: Based on anthropological data, the average height of an adult South American (male or female) is approximately 5.5 feet (1.67 meters). The average shoulder width is about 1.5 feet (0.46 meters), and the body depth (front to back) is about 0.8 feet (0.24 meters).
- Body Orientation: Bodies are stacked horizontally, roughly parallel to the ground, in a somewhat orderly fashion to maximize density, but with some randomness to account for cascading. The pile is assumed to be roughly circular or elliptical at the base to simplify calculations, with a base area matching the given dimensions.
- Pile Shape: The pile is modeled as a cone or pyramid with a base area of 19 feet by 21 feet. The bodies cascade outward, so the pile tapers from the center to the edges, forming a sloped structure.
- Packing Density: Bodies are not perfectly packed due to irregular shapes and cascading. A packing efficiency of about 60% is assumed, meaning 60% of the pile's volume is occupied by bodies, with the rest being air gaps.
- Base Area: The pile's base is 19 feet long and 21 feet wide, giving an area of 19×21=399 19 \times 21 = 399 19×21=399 square feet. For a conical model, we approximate the base as circular with an equivalent area. The radius r r r of a circle with area 399 square feet is:πr2=399 ⟹ r2=399π≈127 ⟹ r≈127≈11.3 feet.\pi r^2 = 399 \implies r^2 = \frac{399}{\pi} \approx 127 \implies r \approx \sqrt{127} \approx 11.3 \text{ feet}.πr2=399⟹r2=π399≈127⟹r≈127≈11.3 feet.
Step 1: Volume of the Pile
The pile's volume depends on its height, which we need to estimate. A conical pile has a volume given by:
V=13πr2h,V = \frac{1}{3} \pi r^2 h,V=31πr2h,
where r≈11.3 r \approx 11.3 r≈11.3 feet and h h h is the height. However, the height depends on how many bodies are stacked and how they cascade. To estimate, we first calculate the number of bodies needed to cover the base and then consider how stacking affects height.
Step 2: Bodies in the Base Layer
Assume bodies are laid flat, side by side, to cover the base area of 399 square feet. The cross-sectional area of one body (lying flat, viewed from above) is approximately:
Body area=length×width=5.5 feet×1.5 feet=8.25 square feet.\text{Body area} = \text{length} \times \text{width} = 5.5 \text{ feet} \times 1.5 \text{ feet} = 8.25 \text{ square feet}.Body area=length×width=5.5 feet×1.5 feet=8.25 square feet.
With a packing efficiency of 60%, the effective area per body is:
8.250.6≈13.75 square feet.\frac{8.25}{0.6} \approx 13.75 \text{ square feet}.0.68.25≈13.75 square feet.
The number of bodies needed to cover the base layer is:
39913.75≈29 bodies.\frac{399}{13.75} \approx 29 \text{ bodies}.13.75399≈29 bodies.
Step 3: Total Number of Bodies
To form a pile, bodies are stacked upward, with each layer having fewer bodies as the pile tapers toward the top. For a conical pile, the number of bodies depends on the volume. The volume occupied by one body is approximately:
Body volume=length×width×depth=5.5×1.5×0.8=6.6 cubic feet.\text{Body volume} = \text{length} \times \text{width} \times \text{depth} = 5.5 \times 1.5 \times 0.8 = 6.6 \text{ cubic feet}.Body volume=length×width×depth=5.5×1.5×0.8=6.6 cubic feet.
With 60% packing efficiency, the effective volume per body is:
6.60.6=11 cubic feet.\frac{6.6}{0.6} = 11 \text{ cubic feet}.0.66.6=11 cubic feet.
To estimate the total number of bodies, we need the pile's volume, which requires the height. The height depends on the slope of the pile, which is determined by how bodies cascade. A natural angle of repose for irregularly stacked objects is typically 30–45 degrees. Let’s assume a 45-degree slope for simplicity, meaning the pile’s height h h h is roughly equal to the radius of the base (r≈11.3 r \approx 11.3 r≈11.3 feet) for a cone with a 45-degree slope.
For a cone with radius r=11.3 r = 11.3 r=11.3 feet and height h=11.3 h = 11.3 h=11.3 feet:
V=13π(11.3)2(11.3)≈13π(127.69)(11.3)≈13⋅400.9⋅11.3≈1510 cubic feet.V = \frac{1}{3} \pi (11.3)^2 (11.3) \approx \frac{1}{3} \pi (127.69) (11.3) \approx \frac{1}{3} \cdot 400.9 \cdot 11.3 \approx 1510 \text{ cubic feet}.V=31π(11.3)2(11.3)≈31π(127.69)(11.3)≈31⋅400.9⋅11.3≈1510 cubic feet.
The number of bodies is:
151011≈137 bodies.\frac{1510}{11} \approx 137 \text{ bodies}.111510≈137 bodies.
Step 4: Height of the Pile
The height of the pile is approximately 11.3 feet, based on the conical model with a 45-degree slope. However, if the pile is more pyramid-like or the slope is shallower (e.g., 30 degrees), the height would be lower, and the number of bodies would adjust accordingly. For a 30-degree slope, the height is roughly:
h=r⋅tan(30∘)≈11.3⋅0.577≈6.5 feet.h = r \cdot \tan(30^\circ) \approx 11.3 \cdot 0.577 \approx 6.5 \text{ feet}.h=r⋅tan(30∘)≈11.3⋅0.577≈6.5 feet.
Recalculating the volume for a cone with h=6.5 h = 6.5 h=6.5 feet:
V = \frac{1}{3} \pi (11.3)^2 (6.5) \approx \fracDonna}{3} \cdot 400.9 \cdot 6.5 \approx 869 \text{ cubic feet}.
Number of bodies:
86911≈79 bodies.\frac{869}{11} \approx 79 \text{ bodies}.11869≈79 bodies.
Final Answer
For a pile 19 feet long and 21 feet wide, with bodies stacked in the middle and cascading down:
- Number of bodies: Approximately 80–140, depending on the slope (30–45 degrees). A reasonable estimate is ~100 bodies for an intermediate slope.
- Height of the pile: Approximately 6.5–11.3 feet. A reasonable estimate is ~9 feet for an intermediate slope.
These are rough estimates, as actual stacking would be highly irregular, and factors like compression or body positioning could vary the results.
