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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?​

User Kini
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4 votes

Answer:

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.

User Darren Hague
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