The vault shown is in the shape of a cube. If each side is \( 4 y^{5} \) feet, find its volume.

To find the volume of a cube, you use the formula \( V = s^3 \), where \( s \) is the length of a side of the cube. In this case, each side is \( 4y^5 \) feet. Calculating the volume: \[ V = (4y^5)^3 = 4^3 \cdot (y^5)^3 = 64y^{15} \] So, the volume of the vault is \( 64y^{15} \) cubic feet. Now let's talk about the significance of volumetric calculations and practical contexts where such shapes appear! Cubical objects, like vaults, can be found in architecture and design. Understanding their volume is essential, especially when it comes to planning storage or ensuring the structural integrity of a building. For example, knowledge of volume helps determine how much air or heat a space can contain, important for climate control in large buildings. Another fun fact is that similar calculations are not limited to cubes! In real-world applications, volume calculations extend to various shapes, including cylinders and spheres. Mistakes often occur when mixing up formulas (like using surface area instead of volume), so a tip is to always double-check the shape and the corresponding formula you’re using. Remember, a little mistake in the formula can lead to a big difference when it comes to actual volume calculations!