Calculating the Rise in Water Level When a Cube is Immersed in a Cuboid Vessel

When a cube is immersed in a cuboid vessel filled with water, the water level rises due to the volume of the cube. How can we calculate the precise rise in the water level? Let's walk through the steps to find the solution.

Introduction to the Problem

The problem involves a cube with an edge length of 6 cm and a cuboid vessel with a base dimension of 12 cm by 10 cm. When the cube is completely immersed, the water does not overflow.

Step 1: Calculate the Volume of the Cube

The volume V of a cube with edge length a is given by the formula:

V a3

Given that a 6 cm, the volume of the cube is: V 63 216 cm3

Step 2: Calculate the Base Area of the Cuboid Vessel

The base area A of the cuboid vessel can be calculated using the dimensions of the base:

A 12 cm × 10 cm 120 cm2

Step 3: Calculate the Rise in Water Level

The rise in water level h can be found using the formula:

h V / A

Substituting the values we calculated:

h 216 cm3 / 120 cm2 1.8 cm

Conclusion

The rise in the water level when the cube is immersed in the cuboid vessel is 1.8 cm.

Alternative Method

Let the initial water height be h. The volume of water initially in the vessel is:

12 cm × 10 cm × h 120h cm3

The volume of the cube is:

63 216 cm3

After the cube is immersed, the total volume of water is:

120h 216 cm3

The maximum volume the vessel can hold is 120 cm × H (where H is the height of the vessel). If the volume of the vessel after the cube is immersed is full but does not overflow, then:

120H 120h 216

Rearranging the equation:

H - h 216 / 120 1.8 cm

This confirms that the rise in the water level is 1.8 cm.

Final Calculation with Vessel Height

Assuming the vessel height is H 10 cm, the initial water height h can be calculated:

120h 10 cm × 120 cm2 × h 984 cm3

Solving for h:

h 984 / 120 8.2 cm

The final water height is 10 cm, hence the rise in water level is:

10 cm - 8.2 cm 1.8 cm

In conclusion, the rise in the water level when the cube is immersed in the cuboid vessel is 1.8 cm.