The Cylinder and Cone Water Tank
A cylindrical tank has an inner radius of 10 cm and a height of 30 cm. It is filled with water to a height of 18 cm. A solid cone with base radius 6 cm and height 9 cm is slowly lowered into the tank until it rests on the bottom, fully submerged. Water does not overflow. A student predicts the water level will rise by exactly 1 cm. Another student says the rise depends only on the cone's volume and the tank's base area, not on the cone's shape.
1. Compute the rise in water level when the cone is fully submerged. Use $\pi \approx 3.14$ if a decimal is needed.
Read the worked answer
Cone volume: $\dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi (6)^2 (9) = \dfrac{1}{3}\pi \cdot 36 \cdot 9 = 108\pi$ cm$^3$.
Tank base area: $\pi (10)^2 = 100\pi$ cm$^2$.
Rise $= \dfrac{108\pi}{100\pi} = 1.08$ cm.
The water rises 1.08 cm (about 1.08 cm). The first student's prediction of exactly 1 cm is slightly low.
Check overflow: new height $= 18 + 1.08 = 19.08$ cm $< 30$ cm, so no overflow.
2. Evaluate the second student's statement: does the rise depend only on the cone's volume and the tank's base area, not on the cone's shape?
Read the worked answer
When an object is fully submerged, the water level rises by the volume of the object divided by the tank's base area:
$$\text{rise} = \frac{V_{\text{object}}}{A_{\text{base}}}$$
Any two objects with the same volume produce the same rise in the same tank, regardless of shape.
The statement is correct, provided the object is fully submerged and the water does not overflow. The shape only matters insofar as it determines the volume.
3. Suppose instead a sphere of radius 4 cm is submerged. Would the water level rise more or less than for the cone? Show the comparison.
Read the worked answer
Sphere volume: $\dfrac{4}{3}\pi (4)^3 = \dfrac{4}{3}\pi \cdot 64 = \dfrac{256\pi}{3} \approx 85.3\pi$ cm$^3$.
Cone volume was $108\pi$ cm$^3$.
Since $108\pi > \dfrac{256\pi}{3}$ (because $324 > 256$), the cone has the larger volume.
The sphere produces a smaller rise. Sphere rise: $\dfrac{256\pi/3}{100\pi} = \dfrac{256}{300} \approx 0.853$ cm, compared with 1.08 cm for the cone.