The dimensions of a rectangular prism that is completely filled with water are $x - 1, 2 x + 1, a n d 3 x - 1$ ft. A solid cube whose sides have a length of $x + 1$ ft each is submerged in it. What is the remaining amount of water in the prism?

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Solution

Dimensions of the rectangular prism:
$x - 1, 2 x + 1, 3 x - 1$

Volume of a rectangular prism
$=$ Length $\times$ Width $\times$ Height

Volume of a rectangular prism
$=$ $(x - 1)$ $\times$ $(2 x + 1)$ $\times$ $(3 x - 1)$

Amount of water inside the prism $=$ Volume of the prism

Amount of water inside the prism
$= 6 x^{3} - 5 x^{2} - 2 x + 1$

Length of the side of the cube $= x + 1$

Volume of the cube $= S i d e^{3}$
Volume of the cube $= (x + 1)^{3}$
Volume of the cube $= x^{3} + 3 x^{2} + 3 x + 1$

$Amount of water remaining$
$= Volume of prism - Volume of cube$

Amount of water remaining
$= (6 x^{3} - 5 x^{2} - 2 x + 1) - (x^{3} + 3 x^{2} + 3 x + 1)$

Amount of water remaining
$= 6 x^{3} - x^{3} - 5 x^{2} - 3 x^{2} - 2 x - 3 x$

Amount of water remaining
$= 5 x^{3} - 8 x^{2} - 5 x$ cu ft

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