if a rectangular prism has a vloume of 36 cubic centimeters list 4 possible dimensions it could have
2 by 3 by 6
1 by 3 by 12
3 by 3 by 4
2 by 2 by 9
18 by two by one
36 by 1 by 1
1 by 4 by 9
The surface area of a prism of dimensions $a\times b\times c$ is equal to
\[2(ab+bc+ca).\]
To minimize the surface area we believe that we want the side lengths to be as close to equal as possible. Notice that 36 is between the cubes 27 and 64, so the optimal dimensions should be something near $3\times3\times3$ and $4\times4\times4$.
When we factor 36 we get $36=3\cdot3\cdot4$, so we conjecture that the minimum surface area is
\[2(3\cdot3+3\cdot4+4\cdot3)=66.\]
The easiest way to verify our conjecture is by exploring other factorizations. If the dimensions are $a\times b\times 1$ with $ab=36$ then the surface area is
\[2(ab+b\cdot1+1\cdot a)=2(36+a+b)>72>66.\]
The only other cases are $2\times 2\times9$ (which has area 80) and $2\times3\times6$ (which has area 72). Therefore the minimum area is $\boxed{66}$.
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To find possible dimensions for a rectangular prism with a volume of 36 cubic centimeters, we need to consider the factors of 36. Here's how you can find 4 possible dimensions:
Step 1: Find the factors of 36:
The factors of 36 are {1, 2, 3, 4, 6, 9, 12, 18, 36}.
Step 2: Pair the factors to find possible dimensions:
To find the dimensions of a rectangular prism, we need to pair up the factors of 36. Let's try different combinations:
Pair 1: 1 x 36 = 36
Pair 2: 2 x 18 = 36
Pair 3: 3 x 12 = 36
Pair 4: 4 x 9 = 36
These are 4 possible combinations of dimensions for a rectangular prism with a volume of 36 cubic centimeters:
1) Length = 1 cm, Width = 1 cm, Height = 36 cm
2) Length = 2 cm, Width = 2 cm, Height = 18 cm
3) Length = 3 cm, Width = 3 cm, Height = 12 cm
4) Length = 4 cm, Width = 4 cm, Height = 9 cm
Note that there are more possible dimensions, but these are just four examples.