Question

The design for a rectangular solid shipping container has a volume of 24x3+6x2−9x 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 \( h(x) = 3x \) using polynomial long division.

### Step 1: Set up the division
We are dividing:
\[
V(x) = 24x^3 + 6x^2 - 9x
\]
by:
\[
h(x) = 3x
\]

### Step 2: Perform the division
1. Divide the leading term of the volume by the leading term of the height:
\[
\frac{24x^3}{3x} = 8x^2
\]
We will write \( 8x^2 \) as part of the quotient.

2. Multiply \( 8x^2 \) by \( 3x \) and subtract:
\[
(24x^3 + 6x^2 - 9x) - (24x^3) = 6x^2 - 9x
\]

3. Now, divide the new leading term \( 6x^2 \) by \( 3x \):
\[
\frac{6x^2}{3x} = 2x
\]
Add \( 2x \) to the quotient.

4. Multiply \( 2x \) by \( 3x \) and subtract:
\[
(6x^2 - 9x) - (6x^2) = -9x
\]

5. Divide \( -9x \) by \( 3x \):
\[
\frac{-9x}{3x} = -3
\]
Add \(-3\) to the quotient.

6. Multiply \(-3\) by \( 3x \) and subtract:
\[
(-9x) - (-9x) = 0
\]
Thus, we have no remainder.

### Step 3: Write the complete quotient
Combining everything, we find that:
\[
\text{Quotient} = 8x^2 + 2x - 3
\]

### Final result
The cross-sectional area of the container is given by:
\[
\text{cross-sectional area} = 8x^2 + 2x - 3
\]

### Answer
So the answer can be written as:
cross-sectional area = \(8x^2 + 2x - 3\)