If a rectangular prism sides are increased by 2,3,4 cm to make a cube and the difference in the volume of the rectangular prism and the cube is 827cm3, what are the original dimensions of the rectangular prism?
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To solve this problem, we need to set up equations based on the given information and then solve them algebraically. Let's start by setting up the equations.
Let's assume the original dimensions of the rectangular prism are length (L), width (W), and height (H). The volume of the rectangular prism is given by:
Original Volume = L * W * H
According to the problem, when the sides of the prism are increased by 2, 3, and 4 cm, the shape becomes a cube. So, the side length of the cube can be represented as (L + 2) cm, (W + 3) cm, and (H + 4) cm. The volume of a cube is given by:
Cube Volume = (L + 2) * (W + 3) * (H + 4)
The problem states that the difference in volume between the rectangular prism and the cube is 827cm^3. Therefore, we can set up the equation as follows:
Original Volume - Cube Volume = 827
Substituting the equations for the volumes:
L * W * H - (L + 2) * (W + 3) * (H + 4) = 827
Simplifying the equation:
LWH - (L + 2)(W + 3)(H + 4) = 827
Expanding the equation:
LWH - (LW + 3L + 2W + 6)(H + 4) = 827
Distributing in the negative sign:
LWH - LWH - 4LW - 12L - 2HW - 8W - 24 - 6H - 24 = 827
Combining like terms:
-4LW - 12L - 2HW - 8W - 6H - 48 = 827
Rearranging the equation:
-4LW - 2HW - 8W - 12L - 6H - 48 = 827
Simplifying further:
-2LW - W(2H + 8) - 2(6L + 3H + 24) = 827
Now we have the equation in terms of L, W, and H. To solve for the original dimensions, we need more information or conditions.