A pyramid with a square base of side 6 cm and height 9 cm is filled with water. If this water is poured into a cone with a radius of 3 cm, what is the height of the water in the cone?
["Title: How to Calculate the Water Height in a Cone When Filling from a Square-Based Pyramid", "When a pyramid with a square base and water-filled contents is poured into a conical vessel, determining the height of the water inside the cone involves key geometric principles and volume calculations. In this article, we explore a specific case: a pyramid with a base side length of 6 cm and a height of 9 cm filled with water, poured into a cone with a radius of 3 cm. We explain step-by-step how to calculate the resulting height of water in the cone.", "---", "### Step 1: Calculate the Volume of Water in the Pyramid", "The pyramid has a square base of side 6 cm and a height of 9 cm. The volume $ V $ of a pyramid is given by the formula:", "$$\nV = \frac{1}{3} \ imes \ ext{Base Area} \ imes \ ext{Height}\n$$", "The base area is:\n$$\n6 , \ ext{cm} \ imes 6 , \ ext{cm} = 36 , \ ext{cm}^2\n$$", "So, the volume of water is:\n$$\nV = \frac{1}{3} \ imes 36 , \ ext{cm}^2 \ imes 9 , \ ext{cm} = \frac{1}{3} \ imes 324 = 108 , \ ext{cm}^3\n$$", "---", "### Step 2: Use the Volume Formula for a Cone to Find Water Height", "The poured water fills a cone of radius $ r = 3 , \ ext{cm} $. The volume $ V $ of a cone is:", "$$\nV = \frac{1}{3} \pi r^2 h\n$$", "We know the volume is 108 cm³, so:\n$$\n108 = \frac{1}{3} \pi (3)^2 h = \frac{1}{3} \pi \ imes 9 \ imes h = 3\pi h\n$$", "Solving for $ h $:\n$$\nh = \frac{108}{3\pi} = \frac{36}{\pi}\n$$", "Using $ \pi \approx 3.1416 $:\n$$\nh \approx \frac{36}{3.1416} \approx 11.46 , \ ext{cm}\n$$", "---", "### Final Answer", "The height of the water in the cone is approximately 11.46 cm when a square-based pyramid with base 6 cm × 6 cm and height 9 cm, containing 108 cm³ of water, is poured into it.", "---", "### Key Takeaways", "- Pyramid volume = $ \frac{1}{3} \ imes \ ext{base area} \ imes \ ext{height} $\n- Cone volume = $ \frac{1}{3} \pi r^2 h $, solve for height using known volume and radius\n- This problem illustrates how geometric shapes and volume conservation enable accurate predictions in hydrodynamics and engineering design", "Understanding these formulas helps in solving real-world problems involving container filling, storage capacity, and fluid dynamics.", "---", "If you want to calculate the height exactly, use:", "$$\nh = \frac{36}{\pi} \approx 11.46 , \ ext{cm}\n$$", "This precise answer combines geometry and algebra to solve water transfer challenges efficiently."]
