Question

select the correct answer.
two buildings on opposites sides of a highway are ( 3x^3 - x^2 + 7x + 100 ) feet apart. one building is ( 2x^2 + 7x ) feet from the highway. the other building is ( x^3 + 2x^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. ( x^3 + 4x^2 + 7x - 18 )
b. ( 2x^3 + x^2 + 7x + 118 )
c. ( 3x^3 - 3x^2 + 100 )
d. ( 2x^3 - 5x^2 + 118 )

Explanation:

Step1: Define the distance formula

Let the distance between the two buildings be \( D \), the distance of the first building from the highway be \( d_1 = 2x^2 + 7x \), and the distance of the second building from the highway be \( d_2 = x^3 + 2x^2 - 18 \). The total distance between the two buildings is given as \( 3x^3 - x^2 + 7x + 100 \). So, we have the equation \( D + d_1 = 3x^3 - x^2 + 7x + 100 \), and we need to solve for \( D \).

Step2: Rearrange the equation to solve for \( D \)

\( D = (3x^3 - x^2 + 7x + 100) - (2x^2 + 7x) \)

Step3: Simplify the expression

First, distribute the negative sign: \( D = 3x^3 - x^2 + 7x + 100 - 2x^2 - 7x \)
Then, combine like terms:

• For \( x^3 \): \( 3x^3 \)
• For \( x^2 \): \( -x^2 - 2x^2 = -3x^2 \)
• For \( x \): \( 7x - 7x = 0 \)
• Constant term: \( 100 \)

So, \( D = 3x^3 - 3x^2 + 100 \)
Wait, there seems to be a mistake. Wait, the total distance between the two buildings is \( 3x^3 - x^2 + 7x + 100 \), and one building is \( 2x^2 + 7x \) from the highway, the other is \( x^3 + 2x^2 - 18 \) from the highway. Wait, maybe the correct approach is: The distance between the two buildings is the sum of their distances from the highway? Wait, no, they are on opposite sides, so the distance between them is the sum of their distances from the highway. Wait, let's re - read the problem.

"Two buildings on opposites sides of a highway are \( 3x^3 - x^2 + 7x + 100 \) feet apart. One building is \( 2x^2 + 7x \) feet from the highway. The other building is \( x^3 + 2x^2 - 18 \) feet from the highway. What is the standard form of the polynomial representing the width of the highway between the two buildings?" Wait, maybe the width of the highway between them is the distance between the two buildings minus the sum of their distances from the highway? No, if they are on opposite sides, the distance between them is the sum of their distances from the highway. Wait, the problem says "the width of the highway between the two buildings" – maybe I misread. Wait, let's check the answer options. The answer options have \( 2x^3 - 5x^2 + 118 \), etc. Wait, maybe the correct equation is: Let the width of the highway between them be \( W \). Then \( (2x^2 + 7x)+W+(x^3 + 2x^2 - 18)=3x^3 - x^2 + 7x + 100 \)

Now, solve for \( W \):

\( W=(3x^3 - x^2 + 7x + 100)- (2x^2 + 7x)-(x^3 + 2x^2 - 18) \)

Step4: Distribute the negative signs

\( W = 3x^3 - x^2 + 7x + 100 - 2x^2 - 7x - x^3 - 2x^2 + 18 \)

Step5: Combine like terms

• \( x^3 \) terms: \( 3x^3 - x^3=2x^3 \)
• \( x^2 \) terms: \( -x^2-2x^2 - 2x^2=-5x^2 \)
• \( x \) terms: \( 7x - 7x = 0 \)
• Constant terms: \( 100 + 18 = 118 \)

So, \( W = 2x^3 - 5x^2 + 118 \)

Answer:

D. \( 2x^3 - 5x^2 + 118 \)