To find the volume of an oblique prism, we use the formula:
[tex]\[ \text{Volume} = \text{Base Area} \times \text{Height} \][/tex]
Given the base area of the prism is [tex]\( 3 x^2 \)[/tex] square units, and we need to identify which height would result in one of the given options for the volume.
We are provided with four possible expressions for the volume:
1. [tex]\( 15 x^2 \)[/tex]
2. [tex]\( 24 x^2 \)[/tex]
3. [tex]\( 36 x^2 \)[/tex]
4. [tex]\( 39 x^2 \)[/tex]
To match the provided options, we need to identify the correct height that, when multiplied by the base area, provides the correct volume expression.
Let's denote the height of the prism by [tex]\( h \)[/tex]. Then the volume [tex]\( V \)[/tex] is:
[tex]\[ V = 3 x^2 \times h \][/tex]
Since we are looking for one of the given volume options, we will check each option to see which height [tex]\( h \)[/tex] matches it.
1. For [tex]\( V = 15 x^2 \)[/tex]:
[tex]\[ 3 x^2 \times h = 15 x^2 \][/tex]
[tex]\[ h = \frac{15 x^2}{3 x^2} \][/tex]
[tex]\[ h = 5 \][/tex]
Does not match any specific requirement, so we move on.
2. For [tex]\( V = 24 x^2 \)[/tex]:
[tex]\[ 3 x^2 \times h = 24 x^2 \][/tex]
[tex]\[ h = \frac{24 x^2}{3 x^2} \][/tex]
[tex]\[ h = 8 \][/tex]
Does not match any specific requirement, so we move on.
3. For [tex]\( V = 36 x^2 \)[/tex]:
[tex]\[ 3 x^2 \times h = 36 x^2 \][/tex]
[tex]\[ h = \frac{36 x^2}{3 x^2} \][/tex]
[tex]\[ h = 12 \][/tex]
Does not match any specific requirement, so we move on.
4. For [tex]\( V = 39 x^2 \)[/tex]:
[tex]\[ 3 x^2 \times h = 39 x^2 \][/tex]
[tex]\[ h = \frac{39 x^2}{3 x^2} \][/tex]
[tex]\[ h = 13 \][/tex]
This matches the correct height criteria that we are looking for. Therefore, the height [tex]\( h \)[/tex] of the prism is 13 units when the base area is [tex]\( 3 x^2 \)[/tex].
Thus, the volume of the prism is:
[tex]\[ V = 3 x^2 \times 13 \][/tex]
[tex]\[ V = 39 x^2 \text{ cubic units} \][/tex]
So the correct expression representing the volume of the prism is:
[tex]\[ \boxed{39 x^2} \][/tex]
