A rectangular prism has dimensions 4 cm by 5 cm by 6 cm. If each dimension is increased by 50%, what is the new volume?
["Understanding Volume Changes: How Increasing Each Dimension of a Rectangular Prism by 50% Affects the Total Volume", "When studying three-dimensional geometry, understanding how dimensions impact volume is essential. A classic example involves a rectangular prism: a simple shape with length, width, and height. In this article, we explore what happens to volume when the three dimensions of a rectangular prism are increased by 50%.", "### What Is a Rectangular Prism?", "A rectangular prism is a three-dimensional shape with six rectangular faces, where all angles are right angles. Its volume is calculated using the formula:", "[ V = \ ext{length} \ imes \ ext{width} \ imes \ ext{height} ]", "For example, if a rectangular prism has dimensions 4 cm × 5 cm × 6 cm, its original volume is:", "[ V = 4 \ imes 5 \ imes 6 = 120 , \ ext{cm}^3 ]", "### Increasing Each Dimension by 50%", "Increasing a dimension by 50% means multiplying it by 1.5 (or 150%). Let’s apply this to each dimension:", "- Original length: 4 cm → 4 × 1.5 = 6 cm\n- Original width: 5 cm → 5 × 1.5 = 7.5 cm\n- Original height: 6 cm → 6 × 1.5 = 9 cm", "### New Volume Calculation", "Now, compute the volume with the new dimensions:", "[ V_{\ ext{new}} = 6 \ imes 7.5 \ imes 9 ]", "First calculate ( 6 \ imes 7.5 ):", "[ 6 \ imes 7.5 = 45 ]", "Then multiply by 9:", "[ 45 \ imes 9 = 405 , \ ext{cm}^3 ]", "### Why Does the Volume Increase So Significantly?", "The volume scales by the product of the scaling factors. Since each dimension increases by 1.5×, the total volume increases by ( 1.5 \ imes 1.5 \ imes 1.5 = 3.375 ) times.", "[ 120 \ imes 3.375 = 405 , \ ext{cm}^3 ]", "This demonstrates how small percentage increases in each dimension lead to substantial volume growth—especially in three-dimensional objects.", "### Conclusion", "If each dimension of a rectangular prism measuring 4 cm × 5 cm × 6 cm is increased by 50%, the new volume becomes 405 cm³. This demonstrates the powerful effect of dimensional scaling on 3D space and is a valuable concept in math, engineering, and design.", "---", "Keywords: rectangular prism volume, 4 cm by 5 cm by 6 cm, volume increase, geometric scaling, 3D geometry, math tutorial, volume calculation, scaling dimensions"]
