Answered

Two buildings on opposite sides of a highway are $3 x^3-x^2+7 x+100$ feet apart. One building is $2 x^2+7 x$ feet from the highway. The other building is $x^3+2 x^2-18$ feet from the highway. What is the standard form of the polynomial representing the width of the highway between the two buildings?


A$2 x^3+x^2+7 x+118$
B$3 x^3-3 x^2+100$
C$2 x^3-5 x^2+118$
D$x^3+4 x^2+7 x-18$

Verified Answer :

  • Identify the total distance polynomial and the individual building distance polynomials.
  • Calculate the combined distance of the buildings by summing their individual distance polynomials: \(S(x) = x^3 + 4x^2 + 7x - 18\).
  • Determine the highway width by subtracting this sum from the total distance polynomial: \(W(x) = (3x^3 - x^2 + 7x + 100) - (x^3 + 4x^2 + 7x - 18)\).
  • Simplify the resulting expression to get the standard form: \(\boxed{2x^3 - 5x^2 + 118}\).

Explanation

  1. Analyzing the Problem

The problem asks us to determine the width of a highway that lies between two buildings. We are given:

  1. The total distance between the two buildings, given by the polynomial \(D(x) = 3x^3 - x^2 + 7x + 100\).
  2. The distance of the first building from the highway, \(d_1(x) = 2x^2 + 7x\).
  3. The distance of the second building from the highway, \(d_2(x) = x^3 + 2x^2 - 18\).

Since the buildings are on opposite sides of the highway, the highway width \(W(x)\) is the difference between the total distance and the sum of the distances of the two buildings from the highway. Thus, we have the relationship \(W(x) = D(x) - (d_1(x) + d_2(x))\).

  1. Calculating the Sum of Distances

First, we find the sum of the distances of the two buildings from the highway:
\[S(x) = d_1(x) + d_2(x) = (2x^2 + 7x) + (x^3 + 2x^2 - 18).\]
By grouping like terms, we get:
\[S(x) = x^3 + (2x^2 + 2x^2) + 7x - 18 = x^3 + 4x^2 + 7x - 18.\]

  1. Finding the Highway Width Polynomial

Now, we subtract this sum from the total distance \(D(x)\):
\[W(x) = D(x) - S(x) = (3x^3 - x^2 + 7x + 100) - (x^3 + 4x^2 + 7x - 18).\]
Distributing the negative sign:
\[W(x) = 3x^3 - x^2 + 7x + 100 - x^3 - 4x^2 - 7x + 18.\]
Grouping like terms:
\[W(x) = (3x^3 - x^3) + (-x^2 - 4x^2) + (7x - 7x) + (100 + 18).\]
Simplifying:
\[W(x) = 2x^3 - 5x^2 + 118.\]

  1. Final Conclusion

Comparing our result \(2x^3 - 5x^2 + 118\) to the given options, we identify the correct answer as the third option.
The width of the highway is represented by the polynomial \(\boxed{2x^3 - 5x^2 + 118}\).

Examples

In urban planning, calculating the width of a corridor or highway between structures is a classic application of polynomial operations. By representing distances as polynomials based on factors like terrain slope (\(x\)), engineers can subtract structure offsets from total plot widths to ensure zoning compliance and safety buffers are maintained. This helps determine available space for road expansion or utility placement in a consistent and mathematical way.

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