Question

Monica found the volume of the prism below using the expression 4 + (4 times one-half).

A prism has a length of 2 centimeters, height of 1 and two-thirds centimeters, and width of 2 centimeters.

What errors did Monica make? Check all that apply.
She should have counted only 2 unit cubes instead of 4.
She should have counted 8 one-third cubes instead of only 4.
She should have found the total value of the one-third cubes by multiplying by One-third instead of by One-half.
Each fractional cube is worth Two-thirds instead of the One-half that she used.
She cannot stack more than one layer of fractional cubes in a prism.

Answer 1

To accurately analyze any errors made by Monica in calculating the volume of the prism, we first need to calculate the actual volume of the prism using the given dimensions:

1. **Length** = 2 cm
2. **Height** = \(1 \frac{2}{3} = \frac{5}{3}\) cm
3. **Width** = 2 cm

The volume \(V\) of a prism is computed using the formula:

\[
V = \text{Length} \times \text{Height} \times \text{Width}
\]

Substituting in the values:

\[
V = 2 \times \frac{5}{3} \times 2 = \frac{20}{3} \text{ cm}^3 \approx 6.67 \text{ cm}^3
\]

Now, looking at Monica's expression:

**Monica used**: \(4 + (4 \times \frac{1}{2})\)

This seems to indicate she was adding a volume represented by 4 cubes, which may imply she thought of it in whole units rather than calculating correctly based on dimensions.

Now let's evaluate the potential errors she might have made:

1. **She should have counted only 2 unit cubes instead of 4.**
This statement does not apply. The volume calculation yielded \(\frac{20}{3}\), which cannot be simply represented as unit cubes, especially without knowing how she determined the count of "4" initially.

2. **She should have counted 8 one-third cubes instead of only 4.**
The prism's volume does not break down neatly into cubes of one-third without proper justification for that specific count. Thus, we cannot confirm any direct comparison to count them this way without additional context. This is misleading.

3. **She should have found the total value of the one-third cubes by multiplying by one-third instead of by one-half.**
This suggests that she miscalculated the contribution of those fractional cubes. If she was multiplying by one-half rather than one-third when determining the volume contributed by fractional sections, this indicates a misunderstanding of how to derive volumes from fractional parts.

4. **Each fractional cube is worth two-thirds instead of the one-half that she used.**
Given the actual dimensions, it is possible that a proper interpretation of sections of the height should lead to a proper adjustment of volume calculations. This statement might apply depending on how she calculated each cube's volume representation.

5. **She cannot stack more than one layer of fractional cubes in a prism.**
This statement is incorrect. Fractional volumes can be stacked in calculations. The volume of a prism considers all three dimensions, and multiple layers, regardless of being whole or fractional, are accounted for simultaneously in volume calculations.

Based on this analysis, the relevant error(s) that apply to Monica's situation might include the third one about incorrectly using one-half instead of one-third, and potentially the fourth one about misestimating the value of fractional cube sections depending on the specifics of her calculations. However, the other statements either misinterpret the process or contest the principles of volumetric calculations incorrectly.