Question

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Answer 1

Given Question

Using Euler's Formula, find the unknown.

$\begin{array}{|c|c|c|c|c} \hline \rm Faces&\rm 8&\rm 5&\rm x_{1}& \rm \\ \\ \hline\rm Vertex&\rm x_{2}&\rm 6&\rm 12 & \rm \\ \\ \hline \rm Edges&\rm 12&\rm x_{3}&\rm 30 & \end{array}$

CALCULATIONS

$\pink{\begin{gathered}\large{\sf{{\underline{Formula \: Used - }}}}  \end{gathered}}$

$\pink{\rm :\longmapsto\:\boxed{\tt{ F \: + \: V \: - \: E \: = \: 2 \: }}}$

where

• F is the number of faces

• V the number of vertices

• E the number of edges.

$\red{\large\underline{\sf{Solution-1}}}$

$\rm :\longmapsto\:Faces \: = \: 8 \: \: \: \\ \\ \rm :\longmapsto\:Vertices = x_{2} \\ \\ \rm :\longmapsto\:Edges = 12 \: \: \:$

So, using Euler's Formula, we have

$\rm :\longmapsto\:8 + x_{2} - 12 = 2$

$\rm :\longmapsto\:x_{2} - 4= 2$

$\rm :\longmapsto\:x_{2}= 2 + 4$

$\rm :\longmapsto\:\boxed{\tt{ \: \: x_{2}= 6 \: \: }}$

$\red{\large\underline{\sf{Solution-2}}}$

$\rm :\longmapsto\:Faces \: = \: 5 \: \: \: \\ \\ \rm :\longmapsto\:Vertices = 6 \\ \\ \rm :\longmapsto\:Edges =x_{3} \: \: \:$

So, using Euler's Formula, we get

$\rm :\longmapsto\:5 + 6 - x_{3} = 2$

$\rm :\longmapsto\:11 - x_{3} = 2$

$\rm :\longmapsto\: - x_{3} = 2 - 11$

$\rm :\longmapsto\: - x_{3} = - 9$

$\rm :\longmapsto\:\boxed{\tt{ \: \: x_{3} = 9\: \: }}$

$\red{\large\underline{\sf{Solution-3}}}$

$\rm :\longmapsto\:Faces \: = \: x_{1} \: \: \: \\ \\ \rm :\longmapsto\:Vertices = 12 \\ \\ \rm :\longmapsto\:Edges =30 \: \: \:$

So, using Euler's Formula, we get

$\rm :\longmapsto\: x_{1} + 12 - 30= 2$

$\rm :\longmapsto\: x_{1} - 18= 2$

$\rm :\longmapsto\: x_{1} = 2 + 18$

$\rm :\longmapsto\: \boxed{\tt{ \: \: x_{1} = 20 \: \: }}$

Answer 2

Answer:

Step-by-step explanation:

Formula is F + V - E

so for 1st one

8 + x - 12 = 2

8 + x = 12 + 2

8 + x = 14

x = 14 - 8

x = 6

vertices = 6

for 2nd one

5 + 6 - x = 2

11 - 2 = x

9 = x

edges = 9

for 3rd one

x + 12 - 30 = 2

x + 12 = 2 + 30

x + 12 = 32

x = 32 + 12

x = 44

faces = 44

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