A rectangular prism has dimensions 8 cm by 5 cm by 12 cm. If the length, width, and height are each increased by 20%, what is the new volume?
["Title: How Increasing Dimensions of a Rectangular Prism by 20% Affects Its Volume\nMeta Description: Discover how scaling the dimensions of a rectangular prism by 20% impacts its volume. Learn the new volume calculation for a prism with original dimensions 8 cm × 5 cm × 12 cm.", "---", "### Understanding Volume Changes When Dimensions Increase by 20%", "A rectangular prism’s volume is calculated using the formula:", "[\n\ ext{Volume} = \ ext{length} \ imes \ ext{width} \ imes \ ext{height}\n]", "In this article, we explore what happens to the volume when each dimension is increased by 20%. Specifically, we apply this increase to a prism with original dimensions 8 cm (length), 5 cm (width), and 12 cm (height), then compute the new volume.", "---", "### Step 1: Calculate Original Volume", "Using the original dimensions:", "[\n\ ext{Original Volume} = 8 , \ ext{cm} \ imes 5 , \ ext{cm} \ imes 12 , \ ext{cm} = 480 , \ ext{cm}^3\n]", "---", "### Step 2: Increase Each Dimension by 20%", "A 20% increase means multiplying each dimension by 1.2:", "- New length = (8 \ imes 1.2 = 9.6) cm\n- New width = (5 \ imes 1.2 = 6.0) cm\n- New height = (12 \ imes 1.2 = 14.4) cm", "---", "### Step 3: Calculate the New Volume", "Now multiply the new dimensions:", "[\n\ ext{New Volume} = 9.6 \ imes 6.0 \ imes 14.4\n]", "Break this down step-by-step for clarity:\nFirst, multiply 9.6 × 6.0:", "[\n9.6 \ imes 6.0 = 57.6\n]", "Then multiply the result by 14.4:", "[\n57.6 \ imes 14.4 = 829.44 , \ ext{cm}^3\n]", "---", "### Step 4: Explore Why Volume Increases More Than 20%", "Even though each side grew by 20%, volume increased from 480 cm³ to 829.44 cm³—a growth of over 72.8%, far greater than the linear 20% increase on each dimension. This exponential effect happens because volume depends on the product of three dimensions, amplifying small relative changes across the whole.", "---", "### Final Answer: The New Volume", "After increasing each dimension of the rectangular prism by 20%, the new volume is:", "[\n\boxed{829.44 , \ ext{cm}^3}\n]", "---", "Key Takeaway: Scaling all three dimensions of a 3D shape by the same percentage increases its volume by a factor of ( (1 + r)^3 ), where (r) is the decimal growth rate. In this case, ( (1.2)^3 = 1.728 ), so volume increases by 2.728 times the original—demonstrating the power of multiplicative growth in geometric scaling."]
