Pyramid Volume Calculation
The problem asks for the volume of a pyramid inscribed within a cube.
- Cube edge length, $s = 12 \text{ cm}$.
- The pyramid shares the same base as the cube.
- The pyramid's apex is at the center of the top face.
Base Area Determination
The base of the pyramid is one face of the cube.
Base Area $= s \times s = 12 \text{ cm} \times 12 \text{ cm} = 144 \text{ cm}^2$.
Height Determination
The height ($h$) of the pyramid is the perpendicular distance from the apex to the base. Since the apex is at the center of the top face and the base is the bottom face, the height is equal to the cube's edge length.
Height, $h = s = 12 \text{ cm}$.
Volume Calculation
The formula for the volume ($V$) of a pyramid is:
$V = \frac{1}{3} \times \text{Base Area} \times h$
Substitute the calculated values:
$V = \frac{1}{3} \times 144 \text{ cm}^2 \times 12 \text{ cm}$
$V = \frac{1}{3} \times 1728 \text{ cm}^3$
$V = 576 \text{ cm}^3$
Final Answer
The volume of the inscribed pyramid is $576 \text{ cm}^3$. This matches Option B.
Similar Questions
- The base area of a prism is increased by 25%, while the height remains the same. By what percentage does the volume increase?
- A pyramid with an equilateral triangular base of side 10 cm and height 14 cm is placed on top of a cube with side length 10 cm. Find the total volume of the composite solid.
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Important Questions from 3-D Mensuration
- Find the surface area of a sphere whose diameter is equal to 98 cm.
- A conical vessel has base radius 31 cm and height 45 cm. Water is poured into the vessel until it is $\frac{2}{3}$ full. Find the volume (in cm³) of water in the vessel.
- A solid metallic sphere of radius 10 cm is melted and recast into 125 identical spheres. What is the ratio of the surface area of the original sphere to the total surface area of 6 smaller spheres so formed?
- Find the volume (in cm³) of the largest right circular cone that can be cut out from a cube with an edge of 8 cm. Use $\pi = \frac{22}{7}$
- The volume of a solid cylinder is 5852 cm³ and its height is 38 cm. What is the total surface area of the solid cylinder? (Round your answer to the nearest integer) (Use $\pi = \frac{22}{7}$)
