To find the width of the highway between the two buildings, we need to follow these steps:
1. Expression for the distance between the buildings:
The total distance between the two buildings is given by the polynomial:
[tex]\[
3x^3 - x^2 + 7x + 100
\][/tex]
2. Expression for the distance of Building 1 from the highway:
Building 1 is given to be [tex]\(2x^2 + 7x\)[/tex] feet from the highway:
[tex]\[
2x^2 + 7x
\][/tex]
3. Expression for the distance of Building 2 from the highway:
Building 2 is given to be [tex]\(x^3 + 2x^2 - 18\)[/tex] feet from the highway:
[tex]\[
x^3 + 2x^2 - 18
\][/tex]
4. Determine the total distance of both buildings combined from the highway:
By adding the distances of each building from the highway, we get:
[tex]\[
(2x^2 + 7x) + (x^3 + 2x^2 - 18)
\][/tex]
This simplifies to:
[tex]\[
x^3 + 4x^2 + 7x - 18
\][/tex]
5. Subtraction to find the width of the highway:
The width of the highway is the total distance between the buildings minus the combined distances from the highway:
[tex]\[
(3x^3 - x^2 + 7x + 100) - (x^3 + 4x^2 + 7x - 18)
\][/tex]
6. Simplification:
We subtract the polynomials:
[tex]\[
3x^3 - x^2 + 7x + 100 - x^3 - 4x^2 - 7x + 18
\][/tex]
Combining like terms:
[tex]\[
(3x^3 - x^3) + (-x^2 - 4x^2) + (7x - 7x) + (100 + 18)
\][/tex]
Simplifies to:
[tex]\[
2x^3 - 5x^2 + 118
\][/tex]
Thus, the standard form of the polynomial representing the width of the highway is:
[tex]\[
\boxed{2x^3 - 5x^2 + 118}
\][/tex]
Therefore, the correct answer is D.
