Question
Use the image to answer the question.
An illustration shows a composite figure of two similar sets where a central pentagon is surrounded by a layer of 5 pentagons, one on each side. Two central pentagons are drawn separately with dashed lines. On each side of the central pentagons, another pentagon is drawn using solid lines. Each central pentagon has 5 surrounding pentagons. One pentagon from each set shares a common side.
A net that folds into a dodecahedron is made up of pentagons. If you decompose each pentagon into one triangle and one trapezoid, how many triangles and how many trapezoids are there?
(1 point)
Responses
12 triangles and 12 trapezoids
12 triangles and 12 trapezoids
10 triangles and 10 trapezoids
10 triangles and 10 trapezoids
6 triangles and 6 trapezoids
6 triangles and 6 trapezoids
60 triangles and 0 trapezoids
60 triangles and 0 trapezoids
include explanation on how to solve it.
Explanation:
In a dodecahedron, there are 12 pentagonal faces. If we decompose each pentagon into one triangle and one trapezoid, we will double the number of faces (since each pentagon will now be two faces). Therefore, there will be 24 faces in total.
Since each pentagon has 5 sides, there will be a total of 5 x 12 = 60 sides. Each triangle has 3 sides and each trapezoid has 4 sides, so we can set up the equation:
3x + 4y = 60
where x is the number of triangles and y is the number of trapezoids. We know that there are a total of 24 faces, so we can also set up the equation:
x + y = 24
We can solve for y in the second equation:
y = 24 - x
Then substitute into the first equation:
3x + 4(24 - x) = 60
Simplify and solve for x:
3x + 96 - 4x = 60
-x + 96 = 60
-x = -36
x = 36
So there are 36 triangles. We can substitute this back into the equation y = 24 - x to find y:
y = 24 - 36
y = -12
This doesn't make sense, as we can't have a negative number of trapezoids. Therefore, we made an error in our calculations.
Let's go back to the first equation:
3x + 4y = 60
We know that x and y have to be integers, so we can start by guessing values for x and solving for y. For example, if x = 6:
3(6) + 4y = 60
18 + 4y = 60
4y = 42
y = 10.5
This doesn't work, as y has to be an integer. Let's try x = 12:
3(12) + 4y = 60
36 + 4y = 60
4y = 24
y = 6
This works! So there are 12 triangles and 12 trapezoids. The correct answer is:
12 triangles and 12 trapezoids.