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 container’s cross-sectional area using polynomial division.(1 point) cross-sectional area = ()x^() + ()x - ()

Answer 1

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 \( h(x) = 3x \). The cross-sectional area \( A(x) \) can be expressed as:

\[
A(x) = \frac{V(x)}{h(x)}
\]

Let's perform the polynomial division of \( 24x^3 + 6x^2 - 9x \) by \( 3x \).

1. **Set up the division**:
\[
24x^3 + 6x^2 - 9x \div 3x
\]

2. **Divide the leading term**:
- The leading term of the numerator is \( 24x^3 \) and the leading term of the denominator is \( 3x \).
- \( \frac{24x^3}{3x} = 8x^2 \).

3. **Multiply**:
- Multiply \( 8x^2 \) by the entire divisor \( 3x \):
\[
8x^2 \cdot 3x = 24x^3
\]

4. **Subtract**:
- Subtract \( 24x^3 \) from \( 24x^3 + 6x^2 - 9x \):
\[
(24x^3 + 6x^2 - 9x) - (24x^3) = 6x^2 - 9x
\]

5. **Repeat the process**:
- Now divide \( 6x^2 \) by \( 3x \):
\[
\frac{6x^2}{3x} = 2x
\]
- Multiply \( 2x \) by \( 3x \):
\[
2x \cdot 3x = 6x^2
\]
- Subtract:
\[
(6x^2 - 9x) - (6x^2) = -9x
\]

6. **Repeat again**:
- Now divide \( -9x \) by \( 3x \):
\[
\frac{-9x}{3x} = -3
\]
- Multiply \( -3 \) by \( 3x \):
\[
-3 \cdot 3x = -9x
\]
- Subtract:
\[
-9x - (-9x) = 0
\]

After completing these steps, the result of the polynomial division is:

\[
A(x) = 8x^2 + 2x - 3
\]

Thus, the cross-sectional area of the shipping container is:

\[
\boxed{8x^2 + 2x - 3}
\]