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Difference between revisions of "2009 Grade 8 CEMC Gauss Problems/Problem 7"

(Created page with "==Problem== The number of faces (<math>F</math>), vertices (<math>V</math>), and edges (<math>E</math>) of a polyhedron are related by the equation <math>F + V - E = 2</math>....")
 
(aaaaa)
Line 10: Line 10:
 
We can combine <math>6</math> and <math>8</math> to get:
 
We can combine <math>6</math> and <math>8</math> to get:
  
−
<math>14 - E = 2
+
<math>14 - E = 2</math>
  
−
Adding </math>E<math> to both sides, we get:
+
Adding <math>E</math> to both sides, we get:
  
−
</math>E + 2 = 14<math>
+
<math>E + 2 = 14</math>
  
−
Subtracting </math>2<math> from both sides of the equation, we get:
+
Subtracting <math>2</math> from both sides of the equation, we get:
  
−
</math>E = \boxed {\textbf {(A) } 12}<math>
+
<math>E = \boxed {\textbf {(A) } 12}</math>
  
 
~anabel.disher
 
~anabel.disher
 
==Solution 2==
 
==Solution 2==
−
We can remember that a rectangular prism has </math>6<math> faces, </math>8<math> vertices, and </math>\boxed {\textbf {(A) } 12}$ edges, without doing any calculation.
+
We can remember that a rectangular prism has <math>6</math> faces, <math>8</math> vertices, and <math>\boxed {\textbf {(A) } 12}</math> edges, without doing any calculation.
  
 
~anabel.disher
 
~anabel.disher

Revision as of 22:56, 18 June 2025

Problem

The number of faces ($F$), vertices ($V$), and edges ($E$) of a polyhedron are related by the equation $F + V - E = 2$. If a polyhedron has $6$ faces and $8$ vertices, how many edges does it have?

$\text{ (A) }\ 12 \qquad\text{ (B) }\ 14 \qquad\text{ (C) }\ 16 \qquad\text{ (D) }\ 18 \qquad\text{ (E) }\ 10$

Solution 1

We can use the equation provided in the problem, and plug in $6$ for $F$, and $8$ for $V$:

$6 + 8 - E = 2$

We can combine $6$ and $8$ to get:

$14 - E = 2$

Adding $E$ to both sides, we get:

$E + 2 = 14$

Subtracting $2$ from both sides of the equation, we get:

$E = \boxed {\textbf {(A) } 12}$

~anabel.disher

Solution 2

We can remember that a rectangular prism has $6$ faces, $8$ vertices, and $\boxed {\textbf {(A) } 12}$ edges, without doing any calculation.

~anabel.disher

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