3. square pyramid: volume \( 1,024 \mathrm{~cm}^{3} \);

1. The volume \( V \) of a square pyramid is given by \[ V = \frac{1}{3}a^2h, \] where \( a \) is the length of the base edge and \( h \) is the height. 2. When the volume is known to be \[ 1024\text{ cm}^3, \] we have \[ \frac{1}{3}a^2h = 1024. \] 3. Multiplying both sides by 3 gives \[ a^2h = 3072. \] 4. Since there are two unknowns (\( a \) and \( h \)), without additional information there are infinitely many possibilities for the dimensions. For example, if we choose a convenient value: - Let \( a = 16\text{ cm} \). Then \[ a^2 = 16^2 = 256. \] - Substitute into the equation: \[ 256h = 3072. \] - Solve for \( h \): \[ h = \frac{3072}{256} = 12\text{ cm}. \] 5. Thus, one acceptable set of dimensions is a base edge of \( 16\text{ cm} \) and a height of \( 12\text{ cm} \), which gives \[ \frac{1}{3}(16^2)(12) = \frac{1}{3}(256)(12) = \frac{1}{3}(3072) = 1024\text{ cm}^3. \]