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A storage unit in the shape of a rectangular prism measures 2xft long, x+8ft wide, and x+9ft tall. What are the dimensions of the storage unit, in feet, if its volume is 792ft^(3) ?

User Dbenham
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1 Answer

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Final Answer:

The dimensions of the storage unit are
\(12 \, \text{ft}\) in length, \(4 \, \text{ft}\) in width, and \(5 \, \text{ft}\) in height, ensuring a volume of
\(792 \, \text{ft}^3\). By solving the cubic equation and considering the meaningful root
\(x = 3\), these dimensions fulfill the specified volume requirement.

Step-by-step explanation:

The rectangular prism representing the storage unit is characterized by dimensions expressed in terms of the variable
\( x \). The volume of this prism, crucially important for storage capacity, is specified as
\( 792 \, \text{ft}^3 \) . Setting up the volume equation,
\( 2x * (x+8) * (x+9) = 792 \) , we embark on the process of simplification, ultimately leading to a cubic equation.

Through factoring, we discern two potential solutions for
\( x \), one of which proves relevant in the context of the problem
—\( x = 3 \). The cubic expression is factored into
\( 2(x-3)(x^2 + 20x + 132) = 0 \), elucidating the significance of
\( x = 3 \) as a root.

By substituting this value back into the expressions denoting length, width, and height, we unveil the conclusive dimensions of the storage unit:
\( 12 \, \text{ft} \) in length, \( 4 \, \text{ft} \) in width, and
\( 5 \, \text{ft} \) in height. Consequently, the storage unit, with its dimensions of
\( 12 * 4 * 5 \) feet, satisfies the stipulated volume requirement of
\( 792 \, \text{ft}^3 \), thereby offering an optimal solution for effective storage space.

User Mchouhan
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