Question

2 in. 4 in. 2\frac{1}{4} in. there are 8 one - quarter cubes and 16 unit cubes so (8×\frac{1}{4}) + 16 will give the volume. there are 8 one - quarter cubes and 8 unit cubes so (8×\frac{1}{4}) + 8 will give the volume. there are 8 one - quarter cubes and 16 unit cubes so (8×4) + 16 will give the volume. there are 8 one - quarter cubes and 8 unit cubes so (8×4) + 8 will give the volume.

Explanation:

Step1: Calculate the volume of one - quarter cubes

The volume of a cube is \(V = s^3\). If the side length of the one - quarter cube is \(\frac{1}{4}\) (assuming the unit cube has side length \(1\)), then the volume of one - quarter cube is \(V_{1/4}=(\frac{1}{4})^3=\frac{1}{64}\). But if we consider the number of one - quarter cubes in terms of the side length of the large prism.
The side length \(2\frac{1}{4}=\frac{9}{4}\). If we assume the unit of the small cube is \(\frac{1}{4}\), then the number of \(\frac{1}{4}\) - length segments along the \(2\frac{1}{4}\) side: \(\frac{9/4}{1/4}=9\) (this part might be a mis - interpretation, but re - focusing on the formula structure).
If we consider the formula for the volume of a rectangular prism \(V = l\times w\times h\). Here \(l = 2\frac{1}{4}=\frac{9}{4}\), \(w = 2\), \(h = 4\)
\(V=\frac{9}{4}\times2\times4=18\)
Now, if we consider the composition:
The number of one - quarter cubes: Let's assume the error is in the formula structure. If we consider the formula for the volume of non - unit cubes.
The volume of a cube with side length \(s\) is \(V = s^3\). If we have a prism decomposed into cubes of side length \(\frac{1}{4}\) and unit cubes (\(s = 1\))
The side length \(2\frac{1}{4}=\frac{9}{4}\). The number of \(\frac{1}{4}\) - length segments in \(2\frac{1}{4}\) is \(9\), but if we consider the formula in the options:
The volume of a cube of side length \(\frac{1}{4}\) is \(V_{1/4}=\frac{1}{4}\times\frac{1}{4}\times\frac{1}{4}\), but if we consider the formula as a wrong decomposition (assuming the problem is about adding volumes of two types of cubes)
The volume of a rectangular prism \(V=l\times w\times h\). \(l = 2\frac{1}{4}=\frac{9}{4}\), \(w = 2\), \(h = 4\)
\(V=\frac{9}{4}\times2\times4=18\)
If we consider the formula \((8\times4)+16\):
\(8\times4 = 32\), \(32+16=48\) (wrong)
If we consider \((8\times\frac{1}{4})+16\): \(8\times\frac{1}{4}=2\), \(2 + 16=18\)

Step2: Analyze the options

The volume of a rectangular prism \(V=l\times w\times h\). Here \(l = 2\frac{1}{4}=\frac{9}{4}\), \(w = 2\), \(h = 4\)
\(V=\frac{9}{4}\times2\times4\)
If we assume a wrong decomposition (but following the option logic):
The volume of a cube of side length \(\frac{1}{4}\): If we consider the number of \(\frac{1}{4}\) - cubes as \(8\) (maybe a miscalculation in the problem's figure perception) and unit cubes (\(V = 1\)) as \(16\)
The formula \((8\times\frac{1}{4})+16\) (where \(8\times\frac{1}{4}\) is the volume of the non - unit cubes and \(16\) is the volume of unit cubes) gives the total volume of the prism

Answer:

There are 8 one - quarter cubes and 16 unit cubes so \((8\times\frac{1}{4})+16\) will give the volume.