Problems

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Area of triangles and rectangles.

1. a. What is the area formula for the triangle?
b. One of the problmes on the Rhind papyrus asks: What is the area of a triangle of height 10 khet and a base of 4 khet? Compute the area.

2. a. What is the area formula for a rectangle?
b. One of the papyri has the problem Find the area of a rectangular plot of land measuring 10 khet by 2.. FInd the area.

3. The Egyptians had a slightly different way of writing up solutions.

[line 1] Example of Calculating a rectangle
[line 2] If someone says to you: A rectangle is 12 setjat [in area] [has a breadth 1/2 1/4 [i.e. 3/4] of its length.[Calculate it]
[line 3] Calculate 1/2 1/4 to get 1. The result is 1 1/3
[line 4] Take this 12 setjat 1 1/3 times. The result is 16.
[line 5] Calculate its square root. The result is 4 for its length [and] 1/2 1/4 of it is 3 for the breadth

a. Write down the question as we would ask it.
b. How would we write up the answer if we needed to hand in a detailed solution?

Circles

4. In one of the papyri the computation of the area of a circle leads to the fact that ... the number <math>4(\frac{8}{9})^2 = 3 \frac{13}{81}</math> plays the role of π. How accurate is this approximation of π?

3-Dimensional shapes

5. We compute the volume of a cylinder of radius R and height H to be: Vol = π <math>R^2*H</math>.
Compute the volume of a cylindrical granary of diameter 9 and height 10. (Rhind papyrus nr 41)

6. We compute the volume of a rectangular perallelopiped (a box) of length L, width W and height H to be: Vol = <math> L*W*H</math>.
Compute the volume of a rectangular granary of length 10, breadth 10 and height 10. (Rhind papyrus nr 44)

General

7.

Pyramid Height base
Great Pyramid (Khufu) 146.6 230.3
Khafre 143.5 215.25
Menkaure 66.45 103.4
Userkaf 49 73.3
Sahure 48 78.5
Queen Khentkawes 72 25
Djedkare 52.5 78.75
Djedkare's Queen 21 41
Teti 52.5 78.5
Teti's Queen (Iput) 21 21

Note the interesting difference between the pyramids of the Kings and those of the Queens. It is quite common for the Queens to have a much smaller pyramid. The volume of a pyramid is given by <math>1/3 b^2 h</math>.
(i) Compute the volume the pyramids of Kings like Djedkare and Teti.
(ii) Compute the volumes of their queen's pyramids.
(iii) Given their volume, how much bigger are the king's monuments?