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# Volume question HARD

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A 4" × 6" × 8" rectangular solid is cut by slicing through the midpoints of three adjacent sides as shown. Find the volume of the outlined solid.

Feb 27, 2021

#1
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We can use 3d vectors.  This pyramid is spanned by the vectors (2,0,0), (0,3,0), and (0,0,4).  Taking the determinant gives us a volume of 8.

Feb 27, 2021
#2
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Side lengths are half of each side so 3*2*4=24/2=12

Feb 27, 2021
#3
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The area of the triangular pyramid is $\frac{1}{3} bh.$ We can make the base the top face, and the base has an area of $\frac{3 \cdot 4}{2} = 6.$ The height that is given is $\frac{1}{2} \cdot 4 = 2.$ Now, we can plug these values in, and we get $$V = \frac{1}{3} bh.$$ $$V = \frac{1}{3} (6)(2)$$ $$\boxed{V = 4 \text{ in}^3}.$$

Feb 27, 2021
#4
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Question: If a blue triangle is the base of a pyramid, then what is its height?

Feb 28, 2021