A rectangular prism has dimensions 4 cm, 5 cm, and 6 cm. If each dimension is increased by 50%, what is the new volume?
["How Increasing Dimensions Affects Volume: A Rectangular Prism with 4 cm × 5 cm × 6 cm", "Volume is a fundamental property in geometry, especially when analyzing shapes like rectangular prisms. In this article, we explore how scaling the dimensions of a rectangular prism impacts its volume, using a prism with exact measurements of 4 cm, 5 cm, and 6 cm — and then increasing each dimension by 50%.", "---", "### Understanding the Original Volume", "A rectangular prism’s volume is found by multiplying its length, width, and height:", "[\n\ ext{Volume} = \ ext{length} \ imes \ ext{width} \ imes \ ext{height}\n]", "Given:\nLength = 4 cm,\nWidth = 5 cm,\nHeight = 6 cm", "Original volume:", "[\nV = 4 \ imes 5 \ imes 6 = 120\ \ ext{cm}^3\n]", "---", "### Increasing Each Dimension by 50%", "When each dimension increases by 50%, we multiply each original measurement by 1.5 (which equals +50%).", "New dimensions:\n- New length = (4 \ imes 1.5 = 6) cm\n- New width = (5 \ imes 1.5 = 7.5) cm\n- New height = (6 \ imes 1.5 = 9) cm", "Now calculate the new volume:", "[\nV_{\ ext{new}} = 6 \ imes 7.5 \ imes 9\n]", "Break it down step-by-step:\nFirst, (6 \ imes 7.5 = 45)\nThen, (45 \ imes 9 = 405)", "So, the new volume is 405 cm³", "---", "### Why Volume Increases More Than Dimensions", "A key insight: increasing each linear dimension by a percentage results in the volume increasing by the cube of that percentage increase. Here, a 50% increase in each dimension corresponds to multiplying each side by 1.5.", "[\n1.5^3 = 1.5 \ imes 1.5 \ imes 1.5 = 3.375\n]", "[\n120\ \ ext{cm}^3 \ imes 3.375 = 405\ \ ext{cm}^3\n]", "This confirms the new volume is 3.375 times the original — a powerful illustration of how cubic growth amplifies changes in three-dimensional space.", "---", "### Real-World Application", "Understanding this concept is essential in fields like architecture, packaging design, and manufacturing, where scaling models or products affects not only size but internal capacity and material use.", "---", "### Summary", "- Original volume: 120 cm³\n- After 50% increase per dimension, volume = 405 cm³\n- Volume increases by a factor of 3.375\n- This demonstrates how volume scales with the cube of linear dimensions", "---", "By recognizing how dimensional changes affect volume, you can better plan and analyze projects involving 3D objects — turning abstract geometry into practical knowledge.", "---", "In summary, when each dimension of a rectangular prism measuring 4 cm × 5 cm × 6 cm is increased by 50%, the new volume becomes 405 cm³ — a clear example of cubic scaling in action."]
