To determine the domain of the linear function \( w(x) = -20x + 180 \) that models the amount of water in the aquarium tank over time \( x \), let's analyze the problem step-by-step:
1. Identify the Function and Variables:
- The linear function given is \( w(x) = -20x + 180 \), where:
- \( w(x) \) represents the amount of water in the tank in gallons.
- \( x \) represents the time in hours.
2. Understanding the Problem:
- The tank starts with 180 gallons of water.
- Water is being drained at a rate of 20 gallons per hour.
3. Determine When the Tank is Empty:
- To find the time \( x \) when the tank is empty, we need to solve for \( x \) when \( w(x) = 0 \) (i.e., no water left).
- Set the equation equal to zero:
[tex]\[
-20x + 180 = 0
\][/tex]
- Solve for \( x \):
[tex]\[
-20x = -180
\][/tex]
[tex]\[
x = \frac{180}{20}
\][/tex]
[tex]\[
x = 9
\][/tex]
- This means the tank will be empty after 9 hours.
4. Define the Domain:
- The domain represents the valid values of \( x \) within the context of the problem.
- Since \( x \) cannot be negative (time cannot go backward) and the tank only has water up to 9 hours, \( x \) ranges from \( 0 \) to \( 9 \).
Therefore, the domain of the function \( w(x) = -20x + 180 \) is:
[tex]\[ [0, 9] \][/tex]
The correct answer is:
[tex]\[ [0, 9] \][/tex]
