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]
