To solve this problem, we need to analyze the given equation [tex]\( y = -9x + 500 \)[/tex]. This equation represents the amount of water [tex]\( y \)[/tex] in the tank at any time [tex]\( x \)[/tex] in minutes since the tank began to leak.
Let's examine each term in the equation:
1. Initial Amount of Water:
- When [tex]\( x = 0 \)[/tex] (i.e., at the start, when the leak just began), we substitute [tex]\( x = 0 \)[/tex] into the equation:
[tex]\[
y = -9(0) + 500 = 500
\][/tex]
- This tells us that there were 500 liters of water in the tank when the leak began.
2. Rate of Decrease of Water:
- The coefficient of [tex]\( x \)[/tex] in the equation is -9. This means that the amount of water [tex]\( y \)[/tex] is decreasing by 9 liters per minute.
Now, let's verify each of the given options:
- Option A:
"There were 500 liters in the tank when the leak started, and it is decreasing by 9 liters per minute."
Based on our analysis:
- Initial amount of water when [tex]\( x = 0 \)[/tex] is indeed 500 liters.
- The rate of water decrease is 9 liters per minute.
Therefore, Option A is true.
- Option B:
"There were 9 liters in the tank when the leak started, and it is decreasing by 500 liters per minute."
- Based on our analysis, there were 500 liters initially, not 9 liters.
- The rate of water decrease is 9 liters per minute, not 500 liters per minute.
Option B is false.
- Option C:
"There were 9 liters in the tank when the leak started, and it is increasing by 500 liters per minute."
- Based on our analysis, there were 500 liters initially, not 9 liters.
- The water is decreasing, not increasing, and the rate is 9 liters per minute.
Option C is false.
- Option D:
"There were 500 liters in the tank after 30 minutes, and it is decreasing by 9 liters per minute."
- Let's check the water amount after 30 minutes:
[tex]\[
y = -9(30) + 500 = -270 + 500 = 230
\][/tex]
- After 30 minutes, there will be 230 liters in the tank, not 500 liters.
Therefore, Option D is false.
Since Option A is the only true statement, we conclude that the correct answer is:
A. There were 500 liters in the tank when the leak started, and it is decreasing by 9 liters per minute.
