Answer:
The answer is A (384 units squared)
Step-by-step explanation:
The following figure is a triangular cross-section prism, so the cross-section of the triangular portion from top to bottom is the same. We can determine that the 12 is the perpendicular height of the prism. Since the cross-section is equal, the top and bottom triangles are identical. After visualizing, the base of the prism should look like the drawing attached.
We can use sine or cosine to solve for the unknown equal sides. The angle 45 degrees shown will be our reference angle. Suppose that the unknown side is a value, "x". Since both sides are equal, it doesn't matter which answer I use.
[tex] \sin(45) = \frac{x}{8 \sqrt{2} } [/tex]
[tex] \cos(45) = \frac{x}{8 \sqrt{2} } [/tex]
sin (45) and cos(45) is equal to [Sqrt (2)]/2. Hence, we can find for x.
[tex] \frac{ \sqrt{2} }{2} = \frac{x}{8 \sqrt{2} } [/tex]
[tex]x = \frac{8 \sqrt{2} \times \sqrt{2} }{2} [/tex]
Multiplying square root 2 by square root 2 will give 2.
Hence,
[tex]x = \frac{8 \times 2}{2} = 8[/tex]
Remember that both sides are equal and the base is a right-angled triangle. We can now find the area of the base.
[tex]base \: area = \frac{1}{2} \times 8 \times 8 = 32[/tex]
Now, we can find the volume, which is base area * Perpendicular height.
The volume is:
[tex]32 \times 12 = 384[/tex]
