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)

Answer 1

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

The volume of the container is given by:
\[
V = 24x^3 + 6x^2 - 9x
\]
The height of the container is given by:
\[
h = 3x
\]

To find the cross-sectional area \(A\), we perform the polynomial division of the volume \(V\) by the height \(h\):
\[
A = \frac{V}{h} = \frac{24x^3 + 6x^2 - 9x}{3x}
\]

Now we will perform the polynomial division:

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 divisor \(3x\) by \(8x^2\):
\[
8x^2 \cdot 3x = 24x^3
\]

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

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

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

6. Subtract this from the current polynomial:
\[
(6x^2 - 9x) - 6x^2 = -9x
\]

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

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

9. Subtract this from the current polynomial:
\[
(-9x) - (-9x) = 0
\]

At this point, we have finished the polynomial division, and we obtain:
\[
A = 8x^2 + 2x - 3
\]

Thus, the container’s cross-sectional area is:
\[
\boxed{8x^2 + 2x - 3}
\]