A solid cone (height 120 cm, radius 60 cm) is placed in a right circular cylinder full of water of height 180 cm such that it touches the bottom. The cylinder radius equals that of the cone. Find the volume of water left in the cylinder.
\(\displaystyle 5.04\times10^5\pi\,\text{cm}^3\)
Step 1: Write down the dimensions.
Step 2: Formula for volume of a cylinder.
\(V_{\text{cylinder}} = \pi r^2 h\)
Substitute: \(V_{\text{cylinder}} = \pi (60)^2 (180)\)
\(= \pi (3600)(180) = 648000\pi\,\text{cm}^3\)
Step 3: Formula for volume of a cone.
\(V_{\text{cone}} = \tfrac{1}{3} \pi r^2 h\)
Substitute: \(V_{\text{cone}} = \tfrac{1}{3}\pi (60)^2 (120)\)
\(= \tfrac{1}{3}\pi (3600)(120) = 144000\pi\,\text{cm}^3\)
Step 4: Water displaced by the cone.
The cone is placed inside the cylinder, so it pushes away (displaces) water equal to its own volume.
Displaced water = \(144000\pi\,\text{cm}^3\).
Step 5: Remaining water in the cylinder.
Initial water in cylinder − Volume displaced by cone
\(= 648000\pi - 144000\pi = 504000\pi\,\text{cm}^3\).
Final Answer: \(5.04 \times 10^5 \pi\,\text{cm}^3\).