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:

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