Question

The design for a rectangular solid shipping container has a volume of 24x3+6x2−9x
24
𝑥
3
+
6
𝑥
2
−
9
𝑥
where x represents inches. The box has a height of 3x
3
𝑥
. Find the container’s cross-sectional area using polynomial division.(1 point)
cross-sectional area = $$x+x−

Answer 1

To find the container's cross-sectional area, we'll need to divide the volume polynomial by the height polynomial.

The volume \( V \) of the shipping container is given by:

\[
V = 24x^3 + 6x^2 - 9x
\]

The height \( h \) of the shipping container is given as:

\[
h = 3x
\]

To find the cross-sectional area \( A \), we perform polynomial division by dividing the volume \( V \) by the height \( h \):

\[
A = \frac{V}{h} = \frac{24x^3 + 6x^2 - 9x}{3x}
\]

Now, let's perform the division step-by-step:

1. Divide the leading term of the numerator \( 24x^3 \) by the leading term of the denominator \( 3x \):
\[
\frac{24x^3}{3x} = 8x^2
\]

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

3. Subtract this from the original volume:
\[
(24x^3 + 6x^2 - 9x) - 24x^3 = 6x^2 - 9x
\]

4. Next, divide the leading term \( 6x^2 \) by \( 3x \):
\[
\frac{6x^2}{3x} = 2x
\]

5. Multiply the entire denominator by \( 2x \):
\[
2x \cdot (3x) = 6x^2
\]

6. Subtract again:
\[
(6x^2 - 9x) - 6x^2 = -9x
\]

7. Now, divide the leading term \( -9x \) by \( 3x \):
\[
\frac{-9x}{3x} = -3
\]

8. Multiply the entire denominator by \( -3 \):
\[
-3 \cdot (3x) = -9x
\]

9. Subtracting:
\[
(-9x) - (-9x) = 0
\]

So the division is complete, and the quotient gives us the cross-sectional area \( A \):

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

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

\[
\text{cross-sectional area} = 8x^2 + 2x - 3
\]