How many Scythians were there?

Question

I was doing a maths test yesterday and the last question on the exam was as follows:

$2500$ years ago, a Scythian king called Ariantas ordered every one of his subjects to bring him an arrow head. After he received them all, he smelted the arrowheads into a large bronze bowl that held 600 amphorae and was six fingers in width. Assuming every arrowhead was $1.9 \text{cm}^3$, and six fingers = $10$ cm, how many Scythian's gave arrowheads to Ariantas? (Assume the bowl is a hemisphere and that one amphora equals $39$ liters)

Volume of a hemisphere = $\frac{2}{3} \pi r^3$

Answer 1

The smelted arrowheads are used to produce a hemispherical shell, the inner radius of which is given by the 600 amphorae ≈ 23,400,000 cubic centimeters (since this is the volume the bowl holds). So the inner radius is given by

$$ \frac{2 \pi}{3} r^3 \ = \ 23,400,000 \ \text{cm.}^3 \ \ \Rightarrow \ \ r \ \approx \ 223.4 \ \text{cm.} \ , $$

which is about 7-1/3 feet. (So it is fairly large.)

The thickness of the bowl is the given 10 cm., making the outer radius of the bowl 233.4 cm. The volume of bronze (sorry, I thought it was gold in my comment) used in producing the bowl is then

$$ \frac{2 \pi}{3} (233.4)^3 \ - \ \frac{2 \pi}{3} (223.4)^3 \ \approx \ 3,278,200 \ \text{cm.}^3 \ . $$

If each arrowhead contains 1.9 cubic centimeters of bronze, this requires about 1,725,400 of them.

EDIT -- As a side note to further aid in imagining this "bowl", bronze alloys have densities in the vicinity of 8.2 grams per c.c., so the mass of this object is around 27 metric tons (or roughly 30 English tons).

Answer 2

Let $N$ be the number of Scythians that gave arrowheads to Ariantas.

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Note that a bowl is a solid hemisphere with a smaller hemisphere removed from it. The volume of liquid it can hold is equal to the volume of the smaller of these two hemispheres. The thickness of the bowl is equal to the difference between the radii of the hemispheres.

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Let $r$ be the radius of the smaller hemisphere in centimeters, so that $r + 10$ is the radius of the larger hemisphere.

Then the amount of liquid that the bowl can hold, in cubic centimeters, is equal to $$ \frac{2}{3} \pi r^3$$

But we know that the bowl can hold $600$ amphorae; we just need to convert that to cubic centimeters and equate that to the volume we found in terms of $r$.

$1$ amphora is $39$ liters, and $1$ liter is $1000$ cubic centimeters, so $600$ amphorae, in $\text{cm}^3$, is $$600 \cdot 39 \cdot 1000 = 23400000$$

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Now we solve for $r$ in centimeters:

$$ \frac{2}{3} \pi r^3 = 23400000$$

$$ r^3 = \frac{35100000}{\pi}$$

$$r = \sqrt[3]{\frac{35100000}{\pi}} \approx 223.556$$

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We need to find the volume of the bowl itself; this is the volume of the larger hemisphere (with radius approximately 233.6) minus the volume of the smaller hemisphere (with radius approximately 223.6):

$$ \frac{2}{3} \pi (r+10)^3 - \frac{2}{3} \pi r^3 $$

$$\approx \left(\frac{2}{3}\pi\right) \cdot \left((233.556)^3 - (223.556)^3)\right)\approx 3282724.209$$

The number of Scythians, $N$, is equal to this volume, divided by the volume of each arrowhead:

$$N \approx \frac{3282724.209}{1.9} \approx \boxed{1,727,750 \text{ Scythians}}$$

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World population estimates between 1000 BCE and 200 CE are about 50 to 300 million, so it is believable that Scythia could have almost 2 million of these people.