Question

which statements about this prism partially packed with unit cubes are true? check all that apply

the middle layer of cubes is missing 7 unit cubes.
there are 9 unit cubes stacked in the prism.
there are 7 unit cubes that are hidden from view inside the prism.
the volume of the prism can be found by ( 12\times3 ) because there are 12 cubes in each layer and there are 3 layers
of cubes.
the volume of the prism can be found by doubling the number of cubes in the prism because it is half full

mark this and return

Explanation:

Step1: Calculate the number of cubes in each layer

The number of cubes in each layer is \(4\times3 = 12\) (using the formula for the area of a rectangle \(A = l\times w\), where \(l = 4\) and \(w = 3\)).

Step2: Analyze each statement

• For the first statement:

The middle - layer: total cubes per layer \(12\). Visible cubes in the middle - layer: \(5\). Missing cubes in the middle - layer \(12-(2 + 3)=7\).

• For the second statement:

Count the visible cubes:
Bottom - layer: \(6\) cubes, middle - layer: \(5\) cubes, top - layer: \(4\) cubes. Total visible cubes \(6 + 5+4=15\). There are also hidden cubes. So, the statement “There are 9 unit cubes stacked in the prism” is false.

• For the third statement:

Total cubes in the prism \(V=4\times3\times3=36\) (using the formula \(V=l\times w\times h\)). Visible cubes \(15\). Hidden cubes \(36 - 15=21
eq7\). So, the statement “There are 7 unit cubes that are hidden from view inside the prism” is false.

• For the fourth statement:

Since the number of cubes in each layer \(n = 4\times3=12\) and the number of layers \(h = 3\), the volume \(V=n\times h=12\times3\) (because the volume of a rectangular prism with unit - cube side length \(s = 1\) is equal to the number of unit cubes, and \(V=\text{(number of cubes per layer)}\times\text{(number of layers)}\)).

• For the fifth statement:

The prism is not half - full. The total number of cubes \(4\times3\times3 = 36\). If we double the number of existing cubes (which is \(15\)), \(2\times15 = 30
eq36\). So, this statement is false.

Answer:

The first and fourth statements are correct. So, the answer is “The middle layer of cubes is missing 7 unit cubes” and “The volume of the prism can be found by \(12\times3\) because there are 12 cubes in each layer and there are 3 layers of cubes”.