Six identical square pyramids can fill the same volume as a cube with the same base. If the height of the cube is $h$ units, what is true about the height of each pyramid?
A. The height of each pyramid is $\frac{1}{2} h$ units.
B. The height of each pyramid is $\frac{1}{3} h$ units.
C. The height of each pyramid is $\frac{1}{5} h$ units.
D. The height of each pyramid is $h$ units.
Answer (1)
Define the volumes of the cube ( V c ) and the pyramid ( V p ) using their respective formulas.
Establish the relationship between the volumes: 6 V p = V c .
Substitute the volume formulas into the equation and simplify.
Solve for the height of the pyramid ( h p ) in terms of the height of the cube ( h ): h p = 2 1 h .
Explanation
Problem Analysis Let's analyze the problem. We are given that six identical square pyramids fill the same volume as a cube with the same base. We need to find the relationship between the height of each pyramid and the height of the cube.
Define Variables Let V c be the volume of the cube and V p be the volume of one pyramid. Let b be the side length of the base of the cube (and the pyramids). The height of the cube is h .
Cube Volume The volume of the cube is given by the formula: V c = b 2 h
Pyramid Volume The volume of one pyramid is given by the formula: V p = 3 1 b 2 h p where h p is the height of the pyramid.
Volume Relationship Since six pyramids fill the same volume as the cube, we have: 6 V p = V c
Substitution Substitute the expressions for V c and V p into the equation: 6 ( 3 1 b 2 h p ) = b 2 h
Simplification Simplify the equation: 2 b 2 h p = b 2 h
Solve for pyramid height Divide both sides by 2 b 2 to solve for h p :
h p = 2 1 h
Final Answer Therefore, the height of each pyramid is 2 1 h units.
Examples
Imagine you're designing a set of building blocks where you want to create a cube using identical square pyramids. Knowing that six such pyramids can form a cube allows you to calculate the exact height each pyramid needs to be, relative to the cube's height. This ensures that when the pyramids are combined, they perfectly form the cube, making your building set geometrically accurate and fun to play with.
