Question

A rectangular tank 160 cm long and 80 cm wide is filled with some water. John puts a cube with side 40 cm long into the water. What is the rise in water level if the cube is completely immersed?​

Answer 1

The water level would rise by
`5\; \rm cm`.

Explanation:

Assume that the initial water level is
`h\; \rm cm`.

Before the cube was placed in the tank, the volume of the water was the same as the volume of a rectangular prism with a length of
`160\; \rm cm`, a height of
`80\; \rm cm`, and a height of
`h\; \rm cm`:

`\begin{aligned} & 160\; {\rm cm} * 80\; {\rm cm} * h\; {\rm cm} \\ =\; & (80 * 160) \, h \; \rm cm^(3) \end{aligned}`.

Let the rise in the water level in the tank be
`x\; \rm cm`. The new water level would be
`(x + h)\; \rm cm`.

The volume of the water and the submerged cube, combined, would be the same as that of a rectangular prism with a length of
`160\; \rm cm`, a width of
`\rm 80\; \rm cm`, and a height of
`(x + h)\; \rm cm`:

`\begin{aligned} & 160\; {\rm cm} * 80\; {\rm cm} * (x + h)\; {\rm cm} \\ =\; & (80 * 160) \, (x + h) \; \rm cm^(3) \end{aligned}`.

In other words:

• Volume of water in this tank:
  `(80 * 160)\, h\; \rm cm^(3)`, whereas
• Volume of water in this tank, plus the volume of the submerged cube:
  `(80 * 160) \, (x + h) \; \rm cm^(3)`.

Therefore, the volume of the submerged cube could be expressed as:

`\begin{aligned} & (80 * 160) \, (x + h) \; {\rm cm^(3)} - (80 * 160) \, h \; \rm cm^(3) \\ =\; &(80 * 160)\, x\; {\rm cm^(3)}\end{aligned}`.

On the other hand, the volume of the cube could be expressed as:

`(40\; \rm cm)^(3) = 64000\; \rm cm^(3)`.

Equate these two expressions for the volume of the cube and solve for
`x`, the rise in the water level in the tank:

`(80 * 160)\, x = 64000`.

`x = 5`.

In other words, the rise in the water level in this tank would be
`5\; \rm cm`.