Question

The design for a rectangular solid shipping container has a volume of 24x^3+6x^2-9x where x represents inches. the box has a height of 3x. Find the containers cross sectional area using polynomial division

Answer 1

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:

1. **Write the volume and height**:
- Volume: \( V = 24x^3 + 6x^2 - 9x \)
- Height: \( h = 3x \)

2. **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.