Question

 A cube of 9 cm edge is immersed completely in a rectangular vessel containing water. If the dimensions of the base are 15 cm and 12 cm, find the rise in water level in the vessel.

Answer 1

Hint: The problem is based on Archimedes' principle. According to the Archimedes principle the volume of the water displaced after immersing a solid in water is equal to the volume of that solid. To solve this question, we will find the volume of the cube and equate it to the volume of the water displaced. This will give us the change in the water level in the vessel.

Complete step by step answer:
We can find the volume of a cube with the formula ${{a}^{3}}$, where a is the length of the side of the cube.
Now, the length of each side of the cube is given as 9 cm.
Therefore, the volume of the cube is ${{\text{V}}_{c}}$ = ${{\left( 9 \right)}^{3}}$ = 729 cubic cm.
The volume of water in a cuboidal vessel is given by the formula V = lbh, where V is the volume, l is the length, b is the breadth and h is the height of the surface of water from the base.
It is given in the question that the dimensions of the base are 15 cm and 12 cm. Let ${{\text{h}}_{\text{1}}}$ be the height of the water surface before immersing the cube.
Therefore, volume of the water will be ${{\text{V}}_{1}}$ = 15(12)(${{\text{h}}_{\text{1}}}$) = 180$ {{\text{h}}_{\text{1}}}$ cubic cm.
Now, after immersing the cube, the height of the water surface will change to ${{\text{h}}_{2}}$.
Therefore, volume of the water now will be ${{\text{V}}_{2}}$ = 15(12)(${{\text{h}}_{2}}$) = 180 ${{\text{h}}_{2}}$ cubic cm
Therefore, change in the volume is given ${{\text{V}}_{2}}-{{\text{V}}_{1}}$
$\Rightarrow {{\text{V}}_{2}}-{{\text{V}}_{1}}=180\left( {{\text{h}}_{2}}-{{\text{h}}_{1}} \right)$ cubic cm
Here, ${{\text{h}}_{2}}-{{\text{h}}_{1}}$ is the rise in the water level in the vessel. Let us denote ${{\text{h}}_{2}}-{{\text{h}}_{1}}$ with r.
Now, according to Archimedes principle, this change in the volume will be equal to the volume of the cube immersed in the water.
\[\begin{align}
  & \Rightarrow {{\text{V}}_{\text{2}}}-{{\text{V}}_{\text{1}}}={{\text{V}}_{c}} \\
 & \Rightarrow 180\left( \text{r} \right)=729 \\
 & \Rightarrow \text{r}=4.05\ \text{cm} \\
\end{align}\]

Note: Students are advised to be vigilant about the dimensions and always write the units in the solution as it will be considered incomplete without the units. Moreover, they are advised to read about some basic physics concepts which will help in various mathematical problems based on the same models.