A box in the shape of a rectangular prism has a volume of 96 cubic feet. The box has a length of (x + 8) feet, a width of x feet, and a height of (x - 2) feet. Find the dimensions... A box in the shape of a rectangular prism has a volume of 96 cubic feet. The box has a length of (x + 8) feet, a width of x feet, and a height of (x - 2) feet. Find the dimensions of the box.
Understand the Problem
The question asks to find the dimensions of a rectangular prism (a box) given its volume and expressions for its length, width, and height in terms of a variable 'x'. We need to set up an equation using the volume formula (Volume = Length * Width * Height), solve for 'x', and then substitute 'x' back into the expressions for length, width, and height to find the numerical dimensions.
Answer
Length = 12 feet, Width = 4 feet, Height = 2 feet
Answer for screen readers
Length = 12 feet Width = 4 feet Height = 2 feet
Steps to Solve
- Write down the volume formula
The volume $V$ of a rectangular prism is given by $V = lwh$, where $l$ is the length, $w$ is the width, and $h$ is the height.
- Set up the equation
Substitute the given expressions for length, width, and height, and the given volume into the formula:
$96 = (x + 8)(x)(x - 2)$
- Expand the equation
Expand the right side of the equation: $96 = (x + 8)(x^2 - 2x)$ $96 = x^3 - 2x^2 + 8x^2 - 16x$ $96 = x^3 + 6x^2 - 16x$
- Rearrange the equation
Move all terms to one side to set the equation to zero: $x^3 + 6x^2 - 16x - 96 = 0$
- Solve for x
We need to find a value of $x$ that satisfies the equation $x^3 + 6x^2 - 16x - 96 = 0$. We can try integer values that are factors of 96. Trying $x = 4$: $4^3 + 6(4^2) - 16(4) - 96 = 64 + 96 - 64 - 96 = 0$
Since $x = 4$ satisfies the cubic equation, $(x - 4)$ is a factor. Now we perform polynomial division or synthetic division to find the remaining quadratic factor. By synthetic division:
4 | 1 6 -16 -96
| 4 40 96
-------------------
1 10 24 0
This gives us the quadratic factor $x^2 + 10x + 24$.
- Factor the quadratic
Factor the quadratic: $x^2 + 10x + 24 = (x + 6)(x + 4)$
- Find all possible solutions for x
So, the solutions for $x$ are $x = 4$, $x = -6$, and $x = -4$.
- Determine the valid solution
Since the width of the box is $x$, $x$ must be positive. Also, the height is $x - 2$, so $x$ must be greater than 2. Therefore, $x = 4$ is the only valid solution.
- Find the dimensions of the box
Substitute $x = 4$ into the expressions for length, width, and height: Length = $x + 8 = 4 + 8 = 12$ feet Width = $x = 4$ feet Height = $x - 2 = 4 - 2 = 2$ feet
Length = 12 feet Width = 4 feet Height = 2 feet
More Information
The dimensions of the rectangular prism are 12 feet (length), 4 feet (width), and 2 feet (height). We can verify that the volume is indeed 96 cubic feet: $12 \times 4 \times 2 = 96$.
Tips
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Not checking for valid solutions: It's important to ensure that the solution for $x$ results in positive dimensions for the box. Negative dimensions are not physically possible, so negative solutions for $x$ must be discarded.
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Incorrectly expanding the equation: Make sure to correctly multiply all terms when expanding the equation $(x+8)(x)(x-2)$. A mistake in the expansion will affect the solution for $x$.
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