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Geometry 12
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Polyhedron; Square Pyramid; 5 faces, 5 vertices, 8 edges; convex
Not a Polyhedron
Polyhedron; Triangular Prism; 5 faces, 6 vertices, 9 edges; convex
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Triangle
Circle
Hexagon
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Some sports relie on having very little friction. In biking for example, the smaller the surface area of the tires, the less friction there is. And thus the faster the rider can go.
Draw the top triangle first (for some triangles you may have to count a horizontal space as 2)
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You can find the surface area by adding up the areas of each surface, but if you could use a formula, it would be quicker
All the lateral surfaces are rectanglesArea = bh Add up the areas L = ah + bh + ch + โฆ + dh
L = (a + b + c + โฆ + d)hPerimeter of base = a + b + c + โฆ + dL = Ph
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/\
๐ = 2 3 + 2 4 = 14๐ฟ = 14 7 = 98๐ต = 3 โ 4 = 12
๐ด = 2 12 + 14 7 = 122
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100 = 2๐๐2 + 2๐๐ 5100 = 2๐๐2 + 10๐๐
0 = 2๐๐2 + 10๐๐ โ 1000 = ๐2 + 5๐ โ 15.915
๐ =โ5 ยฑ 52 โ 4 1 โ15.915
2 1
๐ =โ5 ยฑ 88.662
2๐ = 2.2, โ7.2
Only 2.2 makes sense because the radius must be positive
๐ = 2๐22 + 2๐ 2 5๐ = 8๐ + 20๐ = 28๐
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Lateral area is ยฝ because the sides are triangles.
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Base Area
๐ต =1
2๐๐
๐ต =1
25 โ 8 5.5 = 110
โ2 = 5.52 + 4.82
โ = 7.3
๐ = ๐ต +1
2๐โ
๐ = 110 +1
25 โ 8 7.3 = 256
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Looking for lateral area. ๐ฟ = ๐3 15 = 141.4 ๐๐2
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Cut into two prismsTop
๐ = 1 1 1 = 1Bottom
๐ = 3 1 2 = 6Total
๐ = 1 + 6 = 7
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Base Area (front)Find height of triangle
52 + ๐ฅ2 = 102
25 + ๐ฅ2 = 100๐ฅ2 = 75
๐ฅ = 5 3Area=triangle - square
๐ต =1
210 5 3 โ 32
๐ต = 25 3 โ 9 โ 34.301Volume = Bh
๐ = 25 3 โ 9 6 = 150 ๐ โ 54 โ 205.8
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Find volume of washers without holes: V = ฯ ยฝ2 9 = 7.06858Find volume of hole: V = ฯ(3/8)2 9 = 3.97608Find volume of washers with holes: 7.06858 โ 3.97608 = 3.09251Find volume of one washer: 3.09251/150 = 0.02 in3
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๐ต =1
29 5 = 22.5 ๐2
๐ = 22.5 ๐2 8 ๐ = 180 ๐3
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๐ต =1
2๐๐
1
2๐๐๐๐ก๐๐๐ ๐๐๐๐๐ =
1
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360
6= 30
tan 30 =2
๐
๐ =2
tan 30= 3.464
๐ต =1
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๐ =1
341.569 11 = 152.42
tan 40 =๐
5.8๐ = 5.8 โ tan 40 = 4.8668
๐ =1
3๐4.86682 โ 5.8 = 143.86
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You can think about cutting a sphere into many small regular square pyramids.V = 1/3 Bh the area of all the bases is 4ฯr2 and h = r
Volume of box: 4.5 1.5 1.5 = 10.125
Volume of each ball: 4
3๐. 753 = 1.767
Volume of empty space: 10.125 โ 3 1.767 = 4.824
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Areas: 216
600=
9
25=
๐2
๐2
๐
๐=
9
25=
3
5
Cube surface area: ๐ = 6๐2
216 = 6๐2
36 = ๐2
๐ = 6
Volumes: ๐3
๐3=
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๐3=
33
53216
๐3=
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125
27๐3 = 216(125)๐3 = 1000volume of D is 1000
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