Hello, Mars!

A rectangular fish tank has a base 2 feet wide and 3 feet long. When the tank is

partially filled with water, a solid cube with an edge length of 1 foot is placed in the tank.

If no overflow of water from the tank is assumed, by how many inches will the level of

the water rise when the cube becomes submerged?

We can *talk* our way through this problem.

The original volume of water looks like this:

Code:

*-----------*
/ /|
/ / |
/ / |
*-----------* *
| | /
h| | /2
| |/
*-----------*
3

This block of water has: width 2 feet, length 3 feet, height $\displaystyle h$ feet.

The volume is: .$\displaystyle 2\cdot3\cdot h \:=\:6h\text{ ft}^3$

When the one-foot cube is added, the water looks like this:

Code:

*-----------*
/ /|
/ / *
/ / /|
*-----------* / |
d| |/ |
* - - - - - * *
| | /
h| | /2
| |/
* - - - - - *
3

The water has risen to a height of $\displaystyle h+d$ feet.

This volume is: $\displaystyle 6(h+d)\text{ ft}^3$

. . which is 1 cubic foot more than the original volume.

So we have: .$\displaystyle 6(h+d) \:=\:6h + 1 \quad\Rightarrow\quad 6h + 6d\:=\:6h + 1$

. . . . . . . $\displaystyle 6d \:=\:1 \quad\Rightarrow\quad d \:=\:\tfrac{1}{6}\text{ ft}$

Therefore, the water rises 2 inches.