The water usage at a car wash is modeled by the equation W(x) = 5x3 + 9x2 − 14x + 9, where W is the amount of water in cubic feet and x 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 D(x) = x3 + 2x2 + 15, where D is the amount of water in cubic feet and x is time in hours.

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

 C(x) = 5x3 + 7x2 − 14x − 6 C(x) = 4x3 + 7x2 − 14x + 6 C(x) = 4x3 + 7x2 − 14x − 6 C(x) = 5x3 + 7x2 − 14x + 6

To find the function \( C(x) \) that models the water used by the car wash on a shorter day, we need to subtract the water decrease function \( D(x) \) from the original water usage function \( W(x) \).

### Step 1: Identify the functions
- The original water usage function:
  \[
  W(x) = 5x^3 + 9x^2 - 14x + 9
  \]
- The decrease in water usage function:
  \[
  D(x) = x^3 + 2x^2 + 15
  \]

### Step 2: Subtract \( D(x) \) from \( W(x) \)
To find \( C(x) \), subtract \( D(x) \) from \( W(x) \):
\[
C(x) = W(x) - D(x)
\]
\[
C(x) = \left(5x^3 + 9x^2 - 14x + 9\right) - \left(x^3 + 2x^2 + 15\right)
\]

### Step 3: Simplify the expression
Subtract the corresponding terms:
\[
C(x) = (5x^3 - x^3) + (9x^2 - 2x^2) + (-14x - 0x) + (9 - 15)
\]
\[
C(x) = 4x^3 + 7x^2 - 14x - 6
\]

### Final Answer
The function \( C(x) \) that models the water used by the car wash on a shorter day is:
\[
\textbf{C(x) = 4x^3 + 7x^2 - 14x - 6}
\]

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Would you like more details or have any questions?

### Related Questions:
1. How do you determine the effect of closing the car wash early on water usage?
2. Can \( C(x) \) be used to predict water usage if the car wash is open for less than an hour?
3. How does the cubic term \( x^3 \) affect the water usage over time?
4. What is the significance of the constant term in \( C(x) \)?
5. How would you graph \( C(x) \) to visualize water usage on a shorter day?
6. If the car wash closes even earlier, how would that affect the function \( C(x) \)?
7. Can you determine the total water saved over a day by evaluating \( D(x) \) for a specific value of \( x \)?
8. How would a linear decrease in water usage affect the function \( C(x) \)?

### Tip:
When subtracting polynomial functions, align like terms to ensure accurate subtraction, which helps in avoiding errors and simplifies the expression efficiently.

Explore how to model water usage reduction at a car wash by understanding the function C(x). Learn how to subtract the decrease function D(x) from the original water usage function W(x). Detailed step-by-step solution and implications for water conservation during droughts.

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