Erin designs a trophy stand using one-inch cubes. The stand is shaped like a rectangular prism. How many different stands can Erin design using exactly 12 one-inch cubes?
A) 8
B) 3
C) 5
D) 2
My work is 4*3*1, 12*1*1, 6*2*1, 2*2*3
Erin can make 4 different stands. Could someone help me what I am doing wrong. Thank you for your help.
1 * 12.
2 * 6.
3 * 4.
Total = 3 different stands.
add to Henry's:
base square of 4, up 3
base rectangle of 6, up 2
five different prisms
Question is not clear on orientation of the stand (e.g. 12 blocks in a row is different than 12 blocks in a column)
Total = 10 different stands
Length x Width x Height = 12 cu in
1 x 12 x 1 = 12
2 x 6 x 1 = 12
4 x 3 x 1 = 12
1 x 6 x 2 = 12
2 x 3 x 2 = 12
1 x 4 x 3 = 12
2 x 2 x 3 = 12
1 x 3 x 4 = 12
1 x 2 x 6 = 12
1 x 1 x 12 = 12
Well, let's count together and see if we can figure out the correct answer!
From your work, it seems you're considering the different ways the three dimensions of the rectangular prism can be arranged using 12 cubes. However, keep in mind that the order of the dimensions doesn't matter since a 2x3x2 stand will look the same as a 3x2x2 stand.
So, let's review your calculations:
1) 4 * 3 * 1 = 12 stands - Here, you're considering 4 different possibilities for one dimension and 3 different possibilities for another, with the third being fixed at 1. However, this count would only apply if the order mattered, which it doesn't.
2) 12 * 1 * 1 = 12 stands - This calculation assumes that one dimension can take on 12 different possibilities while leaving the other two fixed at 1. However, this count doesn't consider the fact that the order doesn't matter.
3) 6 * 2 * 1 = 12 stands - Similar to the previous calculation, this considers one dimension having 6 different possibilities, the second having 2, and the remaining one being fixed at 1. Nonetheless, the order doesn't affect the outcome.
4) 2 * 2 * 3 = 12 stands - This calculation assumes that one dimension can take on 2 different possibilities, the second also 2 possibilities, and the third 3 possibilities. Yet, let's keep in mind that the order doesn't make a difference.
If we go through all the possibilities and remove duplicates, we can see that there are actually three different ways to arrange 12 one-inch cubes into a rectangular prism. So the correct answer is:
B) 3
Just remember, when you're stacking cubes for a trophy stand, it's no puzzle to be played twice!
To calculate the number of different trophy stands that Erin can design using exactly 12 one-inch cubes, we need to consider all possible combinations of dimensions for the rectangular prism shape.
From your work, you listed four combinations: 4 * 3 * 1, 12 * 1 * 1, 6 * 2 * 1, and 2 * 2 * 3. However, some of these combinations are equivalent and should not be counted separately. Let's break it down step by step:
First, let's list all the possible factors of 12: 1, 2, 3, 4, 6, and 12.
Now, let's consider the three dimensions of the rectangular prism: length, width, and height.
For each possible length, we need to find the number of combinations of width and height that can multiply together to give 12.
Here are the possible combinations:
1) Length = 1:
Width = 1, Height = 12
Width = 2, Height = 6
Width = 3, Height = 4
2) Length = 2:
Width = 1, Height = 6
Width = 2, Height = 3
3) Length = 3:
Width = 1, Height = 4
4) Length = 4:
Width = 1, Height = 3
So, by counting the combinations, we have a total of 5 different stands Erin can design using exactly 12 one-inch cubes. Therefore, the correct answer is C) 5.