The water usage at a car wash is modeled by the equation [tex]W(x) = 3x^3 + 4x^2 - 18x + 4[/tex], where [tex]W[/tex] is the amount of water in cubic feet and [tex]x[/tex] is the number of hours the car wash is open. The owners of the car wash want to cut back their water usage during a drought and decide to close the car wash early two days a week. The amount of decrease in water used is modeled by [tex]D(x) = x^3 + 2x^2 + 15[/tex], where [tex]D[/tex] is the amount of water in cubic feet and [tex]x[/tex] is the time in hours.

Write a function, [tex]C(x)[/tex], to model the water used by the car wash on a shorter day.

A. [tex]C(x) = 2x^3 + 2x^2 - 18x - 11[/tex]
B. [tex]C(x) = 3x^3 + 2x^2 - 18x + 11[/tex]
C. [tex]C(x) = 3x^3 + 2x^2 - 18x - 11[/tex]
D. [tex]C(x) = 2x^3 + 2x^2 - 18x + 11[/tex]



Answer :

To find the function [tex]\( C(x) \)[/tex] that models the water used by the car wash on a shorter day, we need to account for the decrease in water usage. Given:

- The water usage [tex]\( W(x) = 3x^3 + 4x^2 - 18x + 4 \)[/tex]
- The decrease in water used [tex]\( D(x) = x^3 + 2x^2 + 15 \)[/tex]

To determine the water usage on a shorter day, we subtract the decrease in water used [tex]\( D(x) \)[/tex] from the total water usage [tex]\( W(x) \)[/tex]. Therefore, the function [tex]\( C(x) \)[/tex] is given by:
[tex]\[ C(x) = W(x) - D(x) \][/tex]

Substitute the given functions:
[tex]\[ C(x) = (3x^3 + 4x^2 - 18x + 4) - (x^3 + 2x^2 + 15) \][/tex]

Now, we will simplify this expression step-by-step:

1. Distribute the negative sign through the terms in [tex]\( D(x) \)[/tex]:
[tex]\[ C(x) = 3x^3 + 4x^2 - 18x + 4 - x^3 - 2x^2 - 15 \][/tex]

2. Combine like terms:
- Combine the [tex]\( x^3 \)[/tex] terms:
[tex]\[ 3x^3 - x^3 = 2x^3 \][/tex]
- Combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[ 4x^2 - 2x^2 = 2x^2 \][/tex]
- Combine the constant terms:
[tex]\[ 4 - 15 = -11 \][/tex]

So, the simplified expression for [tex]\( C(x) \)[/tex] is:
[tex]\[ C(x) = 2x^3 + 2x^2 - 18x - 11 \][/tex]

Therefore, the function [tex]\( C(x) \)[/tex] is:
[tex]\[ \boxed{C(x) = 2x^3 + 2x^2 - 18x - 11} \][/tex]

Among the given options, the correct function is:
[tex]\[ C(x) = 2x^3 + 2x^2 - 18x - 11 \][/tex]