Answer:
1982.0 ft³
Step-by-step explanation:
The composite figure can be decomposed into a cube and a square pyramid. The volume is the sum of the volumes of these.
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pyramid
The square base of the pyramid is 12 ft on each side, so the diagonal of the base is 12√2 feet. (The diagonal of a square is √2 times the side length.) Then the distance from a corner to the center of the base is half that, or 6√2 feet.
That distance and the height of the pyramid form a right triangle whose hypotenuse is the given 10 ft measure of the length of the edge of a face of the pyramid. Then the height can be found using the Pythagorean theorem:
h² +(6√2)² = 10²
h² = 100 -72 = 28
h = 2√7 ≈ 5.291503 . . . ft
The volume of the pyramid is ...
V = 1/3Bh = 1/3s²h
V = 1/3(12 ft)²(5.2910503 ft) ≈ 253.992 ft³
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cube
The volume of a cube is given by ...
V = s³
V = (12 ft)³ = 1728 ft³
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total volume
The total volume of the figure is the sum of the pyramid volume and the cube volume:
total volume = 253.992 ft³ +1728 ft³ = 1981.992 ft³
The volume of the figure is about 1982.0 ft³.
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Additional comment
Your figure for the volume seems to assume the height of the pyramid is 8 feet. That length is the slant height of one face of the pyramid. It would be the hypotenuse of a right triangle whose other legs are 6 ft and the height of the pyramid. Then we would have ...
h² +6² = 8²
h² = 64 -36 = 28 . . . . as above
The pyramid height is the perpendicular distance from the plane of the base to the peak. It is measured through the middle of the volume, not along one face.
