Difference between revisions of "The Right Triangle and the Pyramids Exploration"

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(New page: 1. Below are some Pythagorean triples. Compute the seqed - in this case b/a - in each case. {| border="1" |- ! Triple (a, b, c) !! Corner Angle !! Seqed (b/a) |- | ( 3, 4, 5) || <math>...)
 
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1. Below are some Pythagorean triples. Compute the seqed - in this case b/a - in each case.
 
1. Below are some Pythagorean triples. Compute the seqed - in this case b/a - in each case.
 
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2. For the following pyramids determine if they could have been constructed using a technique involving right triangles.
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2. For the following pyramids do the following: <br>
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(i) Compute the Seqed if possible. <br>
 +
(ii)Could the pyramid have been constructed using a technique involving right triangles?
  
 
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! Pyramid !! Height !! base !! Seqed of the pyramid !! Slope !! Possible right triangle
 
! Pyramid !! Height !! base !! Seqed of the pyramid !! Slope !! Possible right triangle
 
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| Great Pyramid (Khufu) || 146.6 || 230.3 ||  || 51.8 ||
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| Great Pyramid (Khufu) || 146.6 || 230.3 ||  || 51.8 degrees ||
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|-
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| Khafre || 143.5 || 215.25 || .75 || 53.1 degrees || 3-4-5 Triangle
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|-
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| Menkaure || 66.45 || 103.4 || .79 || 51.3 degrees || Maybe 3-4-5 Triangle?
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|-
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| Userkaf || 49 ||  73.3 ||  || 53 degrees ||  
 
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| Khafre || 143.5 || 215.25 || .75 || 53.1 || 3-4-5 Triangle
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| Sahure || 48 || 78.5 || || 50.5 degrees ||  
 
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| Menkaure || 66.45 || 103.4 || .79 || 51.3 || Maybe 3-4-5 Triangle?
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| Queen Khentkawes || 72 || 25 ||  || 52 degrees ||
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|-
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| Djedkare || 52.5 || 78.75 || || 52 degrees ||
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|-
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| Djedkare's Queen || 21 ||  41 ||  || 62 degrees ||
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|-
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| Teti || 52.5 ||  78.5 ||  || 53.1 degrees ||  
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|-
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| Teti's Queen (Iput) || 21 ||  21 ||  || 63 degrees ||
 
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3. 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 with a much steeper slope. The volume of a pyramid is given by <math>1/3 b^2 h</math>. <br>
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(i) Compute the volume the pyramids of Kings like Djedkare and Teti. <br>
 +
(ii) Compute the volumes of their queen's pyramids. <br>
 +
(iii) Given their volume, how much bigger are the king's monuments?

Revision as of 16:03, 17 November 2008

1. Below are some Pythagorean triples. Compute the seqed - in this case b/a - in each case.

Triple (a, b, c) Corner Angle Seqed (b/a)
( 3, 4, 5) <math>53.1^o</math>
( 5, 12, 13) 67.4<math>^o</math>
( 7, 24, 25) <math>73.7^o</math>
( 8, 15, 17) <math>61.9^o</math>
( 9, 40, 41) <math>77.3^o</math>
(11, 60, 61) <math>79.6^o</math>
(12, 35, 37) <math>71.1^o</math>
(13, 84, 85) <math>81.2^o</math>
(16, 63, 65) <math>75.6^o</math>
(20, 21, 29) <math>46.4^o</math>
(28, 45, 53) 58.1<math>^o</math>
(33, 56, 65) <math>59.5^o</math>
(36, 77, 85) <math>64.9^o</math>
(39, 80, 89) <math>64.0^o</math>
(48, 55, 73) 48.9<math>^o</math>
(65, 72, 97) <math>47.9^o</math>


2. For the following pyramids do the following:
(i) Compute the Seqed if possible.
(ii)Could the pyramid have been constructed using a technique involving right triangles?

Pyramid Height base Seqed of the pyramid Slope Possible right triangle
Great Pyramid (Khufu) 146.6 230.3 51.8 degrees
Khafre 143.5 215.25 .75 53.1 degrees 3-4-5 Triangle
Menkaure 66.45 103.4 .79 51.3 degrees Maybe 3-4-5 Triangle?
Userkaf 49 73.3 53 degrees
Sahure 48 78.5 50.5 degrees
Queen Khentkawes 72 25 52 degrees
Djedkare 52.5 78.75 52 degrees
Djedkare's Queen 21 41 62 degrees
Teti 52.5 78.5 53.1 degrees
Teti's Queen (Iput) 21 21 63 degrees


3. 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 with a much steeper slope. 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?