To find the container's cross-sectional area, we can take the volume of the shipping container and divide it by its height. The volume is given as \( V = 24x^3 + 6x^2 - 9x \), and the height is given as \( h = 3x \).
To find the cross-sectional area \( A \), we use the formula:
\[ A = \frac{V}{h} \]
Substituting the volume and height into the formula gives us:
\[ A = \frac{24x^3 + 6x^2 - 9x}{3x} \]
Now, we can perform polynomial division.
- Divide each term of the numerator by the denominator:
- \( \frac{24x^3}{3x} = 8x^2 \)
- \( \frac{6x^2}{3x} = 2x \)
- \( \frac{-9x}{3x} = -3 \)
So, the division results in:
\[ A = 8x^2 + 2x - 3 \]
Thus, the cross-sectional area of the shipping container is:
\[ A = 8x^2 + 2x - 3 \quad \text{(square inches)} \]
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