Amita decides that she wants the molds to have a square base from originally having a rectangular prism with a volume of 28 cubic centimeters - how many
possible molds can she use
To determine the number of possible molds that Amita can use, we need to consider the dimensions of the rectangular prism first. Let's assume the length, width, and height of the rectangular prism are L, W, and H, respectively.
Given that the volume of the rectangular prism is 28 cubic centimeters, we can set up the equation:
Volume = Length x Width x Height
28 = L x W x H
Since Amita wants the molds to have a square base, we know that the length and width of the rectangular prism should be the same. Let's say the length/width of the square base is "x", and the height remains as "H". So, we have:
L = x
W = x
H = H
Now, substituting these values into the equation for volume:
28 = x x x x H
28 = x^2 x H
We can solve the equation by listing all the possible factors of 28 and checking if any of them are perfect squares:
Factors of 28: 1, 2, 4, 7, 14, 28
Out of these factors, we need to find the ones that can be expressed as the square of an integer. In this case, 4 can be expressed as 2^2, which means its square root is 2. And since x represents both the length and width, x = 2.
Thus, Amita can use molds with a square base of 2 centimeters on each side. So, there is only one possible mold that she can use.
To determine how many possible molds Amita can use, we need to find the dimensions of the rectangular prism with a volume of 28 cubic centimeters and then calculate how many square bases can be formed from those dimensions.
Let's assume the dimensions of the rectangular prism are length (L), width (W), and height (H). Since the base is rectangular, we have:
Volume = Length x Width x Height
Substituting the given volume of 28 cubic centimeters:
28 = L x W x H
To find the possible dimensions, we need to factorize 28. The factors of 28 are: 1, 2, 4, 7, 14, and 28.
Now, we will try to find dimensions that can form a square base. This means the length and width need to be the same. Let's go through the factors and check for the possibilities:
For 1:
1 x 1 x 28 = 28 (not a square base)
1 x 2 x 14 = 28 (not a square base)
For 2:
2 x 2 x 7 = 28 (square base)
For 4:
4 x 4 x 1 = 16 (square base)
For 7:
7 x 7 x 1 = 49 (not a square base)
For 14:
14 x 14 x 1 = 196 (not a square base)
For 28:
28 x 28 x 1 = 784 (not a square base)
From these calculations, we found that there are two possible molds Amita can use: one with dimensions 2 cm x 2 cm x 7 cm and another with dimensions 4 cm x 4 cm x 1 cm.
Therefore, Amita can use two possible molds with a square base.
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