Two people dig cylindrical holes in the earth: one hole is two feet in diameter by two feet deep, and the other hole is one foot in diameter by one foot deep. The big hole is how many times the size of the small hole? is it twice as large, three times as

It might seem like doubling the dimensions of a cylindrical hole would simply double its size, but the mathematics of volume tells a different story. The key lies in the formula for the volume of a cylinder, which is V = πr²h, where 'r' is the radius and 'h' is the height. This calculation determines the total space a cylinder occupies. Notice that the radius is squared, meaning it has a much greater impact on the total volume than the height. This is why a small change in the diameter can lead to a surprisingly large change in the overall size.

Let's apply this to the two holes. The smaller hole has a one-foot diameter (making its radius 0.5 feet) and a one-foot depth. Its volume is π × (0.5)² × 1, which equals 0.25π cubic feet. The larger hole has a two-foot diameter (a radius of 1 foot) and a two-foot depth. Its volume is π × (1)² × 2, which equals 2π cubic feet. When you compare the two volumes, you find that 2π is exactly eight times larger than 0.25π, revealing the significant difference in their capacities.

This principle is incredibly important in the real world, especially in fields like engineering and construction. Accurately calculating the volume of earth to be moved is crucial for budgeting, planning, and ordering materials for any excavation project. Whether digging a foundation for a skyscraper or a trench for a small pipe, engineers must perform these precise calculations. An error could lead to costly delays and material shortages, demonstrating how this simple geometric formula has a massive impact on building our modern world.