Question

The design for a rectangular solid shipping container has a volume of
30x^3+5x^2-15x where x represents inches. The box has a height of 5x. Find the container’s cross-sectional area using polynomial division.

Answer 1

To find the cross-sectional area of the shipping container when given its volume and height, we first need to divide the volume polynomial by the height polynomial.

The volume of the container is given by:

\[
V = 30x^3 + 5x^2 - 15x
\]

The height of the container is given by:

\[
h = 5x
\]

To find the cross-sectional area \( A \), we need to perform the polynomial division of \( V \) by \( h \):

\[
A = \frac{V}{h} = \frac{30x^3 + 5x^2 - 15x}{5x}
\]

We can simplify this division step by step.

1. **Divide the leading term of the numerator by the leading term of the denominator:**

\[
\frac{30x^3}{5x} = 6x^2
\]

2. **Multiply the entire denominator by this result:**

\[
6x^2 \cdot (5x) = 30x^3
\]

3. **Subtract this from the original polynomial:**

\[
(30x^3 + 5x^2 - 15x) - 30x^3 = 5x^2 - 15x
\]

4. **Now, divide the leading term of the new polynomial by the leading term of the denominator:**

\[
\frac{5x^2}{5x} = x
\]

5. **Multiply the entire denominator by this result:**

\[
x \cdot (5x) = 5x^2
\]

6. **Subtract this from the current polynomial:**

\[
(5x^2 - 15x) - 5x^2 = -15x
\]

7. **Next, divide the leading term of the new polynomial by the leading term of the denominator:**

\[
\frac{-15x}{5x} = -3
\]

8. **Multiply the entire denominator by this result:**

\[
-3 \cdot (5x) = -15x
\]

9. **Subtract this from the current polynomial:**

\[
(-15x) - (-15x) = 0
\]

Having completed the polynomial division, we find:

\[
A = 6x^2 + x - 3
\]

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

\[
A = 6x^2 + x - 3 \quad \text{(square inches)}
\]