Question

what does the y-intercept of the line tell you about the situation? the amount of water in the tank decreases by 50 gallons every 5 minutes. the tank has 200 gallons in it when jack opens the valve. the tank will take 20 minutes to completely drain. the tank cannot hold more than 200 gallons.

Explanation:

1. Recall the definition of the y - intercept: In a linear equation \(y = mx + b\) (where \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the slope, and \(b\) is the y - intercept), the y - intercept (\(b\)) represents the value of \(y\) when \(x = 0\).
2. In the context of this problem, the \(x\) - axis represents time (in minutes) and the \(y\) - axis represents the amount of water in the tank (in gallons). When \(x = 0\), it means the time when Jack just opens the valve (time \(= 0\) minutes).
3. Analyze the first option: "The amount of water in the tank decreases by 50 gallons every 5 minutes" describes the rate of change (slope) of the line, not the y - intercept.
4. Analyze the second option: If we consider the linear model of the water in the tank over time, when \(x = 0\) (time when the valve is opened), the y - intercept gives the initial amount of water. From the graph's trend and the concept of y - intercept, if we assume the linear equation, when \(x = 0\), the amount of water is 200 gallons (this can be inferred from the slope calculation and the points on the line. For example, from \(x = 5\) (5 minutes) with \(y = 150\) and \(x = 10\) (10 minutes) with \(y = 100\), the slope \(m=\frac{100 - 150}{10 - 5}=\frac{- 50}{5}=- 10\) gallons per minute. Using the point - slope form \(y - y_1=m(x - x_1)\) with \((x_1,y_1)=(5,150)\), we get \(y-150=-10(x - 5)\). When \(x = 0\), \(y-150=-10(-5)\Rightarrow y - 150 = 50\Rightarrow y=200\)). So this option is about the initial amount of water (y - intercept).
5. Analyze the third option: "The tank will take 20 minutes to completely drain" is about the x - intercept (when \(y = 0\)), not the y - intercept.
6. Analyze the fourth option: "The tank cannot hold more than 200 gallons" is about the tank's capacity, which is not related to the y - intercept (the y - intercept is the initial amount of water, not the maximum capacity).

Answer:

B. The tank has 200 gallons in it when Jack opens the valve.