To find the cross-sectional area of a rectangular solid shipping container given its volume \( V = 24x^3 + 6x^2 - 9x \) and a height \( h = 3x \), we need to divide the volume by the height to get the cross-sectional area.
We can set up the polynomial division as follows:
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Write the volume and height:
- Volume: \( V = 24x^3 + 6x^2 - 9x \)
- Height: \( h = 3x \)
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Perform polynomial long division: We need to divide \( 24x^3 + 6x^2 - 9x \) by \( 3x \).
\[ \text{Division} : \frac{24x^3 + 6x^2 - 9x}{3x} \]
Step 1: Divide the leading term of the numerator by the leading term of the denominator: \[ \frac{24x^3}{3x} = 8x^2 \]
Step 2: Multiply the entire divisor \( 3x \) by \( 8x^2 \): \[ 8x^2 \cdot 3x = 24x^3 \]
Step 3: Subtract this from the original polynomial: \[ (24x^3 + 6x^2 - 9x) - 24x^3 = 6x^2 - 9x \]
Step 4: Now repeat the process with the new polynomial \( 6x^2 - 9x \):
- Divide the leading term: \[ \frac{6x^2}{3x} = 2x \]
Step 5: Multiply the divisor \( 3x \) by \( 2x \): \[ 2x \cdot 3x = 6x^2 \]
Step 6: Subtract this from the new polynomial: \[ (6x^2 - 9x) - 6x^2 = -9x \]
Step 7: Repeat again with \(-9x\):
- Divide the leading term: \[ \frac{-9x}{3x} = -3 \]
Step 8: Multiply the divisor \( 3x \) by \(-3\): \[ -3 \cdot 3x = -9x \]
Step 9: Subtract this from \(-9x\): \[ -9x - (-9x) = 0 \]
At this point, there is no remainder. Thus, the result of the polynomial division is: \[ 8x^2 + 2x - 3 \]
This polynomial represents the cross-sectional area \( A \) of the shipping container when the height is \( 3x \).
Final Answer: The cross-sectional area of the container is \( A = 8x^2 + 2x - 3 \) square inches.