Question

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? the height of each pyramid is \\(\frac{1}{2}h\\) units. the height of each pyramid is \\(\frac{1}{3}h\\) units. the height of each pyramid is \\(\frac{1}{6}h\\) units. the height of each pyramid is h units.

Explanation:

Step1: Recall Volume Formulas

The volume of a cube is \( V_{cube} = s^2h \) (where \( s \) is the side length of the base, and \( h \) is the height). The volume of a square pyramid is \( V_{pyramid} = \frac{1}{3}s^2H \) (where \( H \) is the height of the pyramid, and \( s \) is the side length of the square base).

Step2: Set Up Volume Relationship

We know that six identical square pyramids fill the same volume as the cube. So, \( 6 \times V_{pyramid} = V_{cube} \). Substituting the formulas, we get \( 6 \times \frac{1}{3}s^2H = s^2h \).

Step3: Simplify the Equation

Simplify the left - hand side: \( 6\times\frac{1}{3}s^2H = 2s^2H \). So the equation becomes \( 2s^2H = s^2h \). Since \( s
eq0 \) (the base has a non - zero length), we can divide both sides of the equation by \( s^2 \), getting \( 2H = h \). Then, solve for \( H \): \( H=\frac{1}{2}h \).

Answer:

The height of each pyramid is \(\frac{1}{2}h\) units. (The option: The height of each pyramid is \(\frac{1}{2}h\) units.)