To find the cross-sectional area of the rectangular solid shipping container, we need to divide the volume polynomial \( V(x) = 24x^3 + 6x^2 - 9x \) by the height \( h(x) = 3x \) using polynomial long division.
Step 1: Set up the division
We are dividing: \[ V(x) = 24x^3 + 6x^2 - 9x \] by: \[ h(x) = 3x \]
Step 2: Perform the division
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Divide the leading term of the volume by the leading term of the height: \[ \frac{24x^3}{3x} = 8x^2 \] We will write \( 8x^2 \) as part of the quotient.
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Multiply \( 8x^2 \) by \( 3x \) and subtract: \[ (24x^3 + 6x^2 - 9x) - (24x^3) = 6x^2 - 9x \]
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Now, divide the new leading term \( 6x^2 \) by \( 3x \): \[ \frac{6x^2}{3x} = 2x \] Add \( 2x \) to the quotient.
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Multiply \( 2x \) by \( 3x \) and subtract: \[ (6x^2 - 9x) - (6x^2) = -9x \]
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Divide \( -9x \) by \( 3x \): \[ \frac{-9x}{3x} = -3 \] Add \(-3\) to the quotient.
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Multiply \(-3\) by \( 3x \) and subtract: \[ (-9x) - (-9x) = 0 \] Thus, we have no remainder.
Step 3: Write the complete quotient
Combining everything, we find that: \[ \text{Quotient} = 8x^2 + 2x - 3 \]
Final result
The cross-sectional area of the container is given by: \[ \text{cross-sectional area} = 8x^2 + 2x - 3 \]
Answer
So the answer can be written as: cross-sectional area = \(8x^2 + 2x - 3\)