The Moscow Papyrus

The Moscow papyrus contains only about 25, mostly practical, examples. The author is unknown. It was purchased by V. S. Golenishchev (d. 1947) and sold to the Moscow Museum of Fine Art. Origin:  1700 BC. It is 15 feet long and about 3 inches wide.

Problem 14. Volume of a frustum. The scribe directs one to square the numbers two and four and to add to the sum of these squares the product of two and four. Multiply this by one third of six. "See, it is 56; your have found it correctly."

What the student has been directed to compute is the number

$$V={1\over 3}\cdot 6(2^2+4^2+2\cdot 4)=56$$

Here's the picture that is found in the Moscow Papyrus.

Here's the modern version of the picture and a perspective drawing.

The general formula for a frustum was evidently known to the Egyptians. It is:

$$V={1\over3} h (b_1^2+b_1b_2+b_2^2)$$

Taking $b_1=0$, we get the formula

$$V={1\over3} hb^2$$

This was evidently known also.

Question. Speculate on how the Egyptians could have known the formula for a frustum, given that its derivation depends on the methods of modern calculus.

Problem 10 Compute the surface area of a Quonset type hut roof, which is the earliest estimation of curvilinear area.