To determine the dimensions of the party-favor bag with a volume of 140 cubic inches, we start with the given cubic equation:
[tex]\[ x^3 + 6x^2 - 27x - 140 = 0 \][/tex]
Our goal is to solve this equation for [tex]\( x \)[/tex]. The potential solutions will help us identify the correct dimensions.
By solving the cubic equation:
[tex]\[ x = -7, -4, 5 \][/tex]
These values of [tex]\( x \)[/tex] will represent potential lengths, widths, or heights of the party-favor bag.
Next, we evaluate the following candidate dimensions:
1. Length of 7 inches, Width of 4 inches, and Height of 16 inches.
2. Length of 5 inches, Width of 2 inches, and Height of 14 inches.
3. Length of 4 inches, Width of 1 inch, and Height of 13 inches.
4. Length of 3 inches, Width of 0 inches, and Height of 12 inches (Note: This dimension is not valid as the width must be non-zero).
We calculate the volume for each set of dimensions to check if it meets the requirement of 140 cubic inches:
1. [tex]\((7, 4, 16)\)[/tex]:
[tex]\[
\text{Volume} = 7 \times 4 \times 16 = 448 \text{ cubic inches}
\][/tex]
This does not match 140 cubic inches.
2. [tex]\((5, 2, 14)\)[/tex]:
[tex]\[
\text{Volume} = 5 \times 2 \times 14 = 140 \text{ cubic inches}
\][/tex]
This matches the required volume.
3. [tex]\((4, 1, 13)\)[/tex]:
[tex]\[
\text{Volume} = 4 \times 1 \times 13 = 52 \text{ cubic inches}
\][/tex]
This does not match 140 cubic inches.
Based on the calculations, the only set of dimensions that satisfies the volume requirement of 140 cubic inches is:
[tex]\[
\text{Length} = 5 \text{ inches}, \text{ Width} = 2 \text{ inches}, \text{ Height} = 14 \text{ inches}
\][/tex]
Thus, the dimensions of the party-favor bag are:
[tex]\[
\boxed{5 \text{ inches}, 2 \text{ inches}, 14 \text{ inches}}
\][/tex]
