Question

goran is filling a tank with water. the function graphed below shows the amount of water, v(x) (in liters), in the tank x minutes after he started.
amount of water in the tank (liters)
time (minutes)
find v(6) and interpret this value.
(a) v(6) =
(b) the answer to part (a) means that after minutes there are liters in the tank.

Explanation:

Part (a)

Step1: Identify the graph's coordinates

The graph is a linear function. We can see that at \( x = 0 \) (time = 0 minutes), the volume \( V(0)=240 \) liters. Let's find the slope. Let's take another point, say when \( x = 6 \), we can calculate or from the graph, we can see the pattern. Wait, actually, looking at the grid, each minute, how much does the volume increase? Wait, the y - axis is volume (liters) and x - axis is time (minutes). Let's find two points. At \( x = 0 \), \( V(0)=240 \). Let's see when \( x = 6 \), let's calculate the slope first. Wait, maybe the line goes from (0, 240) to, say, (10, 360)? Wait, no, let's check the grid. Wait, the x - axis is time in minutes (1,2,3,...) and y - axis is volume in liters (30,60,...). Wait, the initial point is (0, 240) (since at x = 0, y = 240). Then, let's see the rate of change. Let's take two points: (0, 240) and (6,?). Wait, maybe the slope is \( \frac{360 - 240}{6 - 0}=\frac{120}{6} = 20 \)? Wait, no, maybe I got the direction wrong. Wait, the graph is increasing, so as time increases, volume increases. Wait, at x = 0, V = 240. At x = 6, let's count the grid. Each small square: x - axis, each unit is 1 minute. y - axis, each unit is 30 liters? Wait, no, the y - axis labels are 0,30,60,90,120,150,180,210,240,270,300,... So each grid line on y is 30 liters. x - axis: each grid line is 1 minute. So at x = 0, V = 240. At x = 6, let's see the line. From (0,240), moving 6 units to the right (x = 6) and how much up? Let's see the slope. Let's take another point, say x = 10, V = 360? Wait, 240 + 120 = 360, over 10 minutes? No, that would be slope 12. Wait, maybe my initial point is wrong. Wait, the graph is labeled "Amount of water in the tank (liters)" on the y - axis (vertical) and "Time (minutes)" on the x - axis (horizontal). Wait, no, actually, the vertical axis is V(x) (volume) and horizontal is x (time). So the graph is a line with x (time) on the horizontal and V(x) (volume) on the vertical. So when x = 0 (time = 0), V(0)=240. Then, let's find the equation of the line. Let's take two points: (0, 240) and (6,?). Wait, looking at the graph, when x = 6, let's see the y - value. Let's count the grid. From x = 0 (240) to x = 6, how much does it go up? Let's see, the line passes through (0, 240) and let's say (6, 240+620)? Wait, maybe the slope is 20 liters per minute. So \( V(x)=240 + 20x \). Then, when x = 6, \( V(6)=240+206=240 + 120 = 360 \). Wait, that makes sense. Let's check with another point. If x = 10, \( V(10)=240 + 2010 = 440 \)? Wait, no, maybe my slope is wrong. Wait, maybe the graph is from (0, 240) to (10, 360)? Then slope is \( \frac{360 - 240}{10 - 0}=\frac{120}{10}=12 \). Then \( V(x)=240 + 12x \). Then at x = 6, \( V(6)=240+126=240 + 72 = 312 \)? No, that doesn't match. Wait, maybe I misread the graph. Wait, the y - axis has labels 0,30,60,90,120,150,180,210,240,270,300,330,360,390,420,... So each major grid line is 30 liters. The x - axis: 0,1,2,3,4,5,6,7,8,9,10,... minutes. The line starts at (0, 240) ([SSE onError error]

Answer:

Step1: Identify the graph's coordinates

The graph is a linear function. We can see that at \( x = 0 \) (time = 0 minutes), the volume \( V(0)=240 \) liters. Let's find the slope. Let's take another point, say when \( x = 6 \), we can calculate or from the graph, we can see the pattern. Wait, actually, looking at the grid, each minute, how much does the volume increase? Wait, the y - axis is volume (liters) and x - axis is time (minutes). Let's find two points. At \( x = 0 \), \( V(0)=240 \). Let's see when \( x = 6 \), let's calculate the slope first. Wait, maybe the line goes from (0, 240) to, say, (10, 360)? Wait, no, let's check the grid. Wait, the x - axis is time in minutes (1,2,3,...) and y - axis is volume in liters (30,60,...). Wait, the initial point is (0, 240) (since at x = 0, y = 240). Then, let's see the rate of change. Let's take two points: (0, 240) and (6,?). Wait, maybe the slope is \( \frac{360 - 240}{6 - 0}=\frac{120}{6} = 20 \)? Wait, no, maybe I got the direction wrong. Wait, the graph is increasing, so as time increases, volume increases. Wait, at x = 0, V = 240. At x = 6, let's count the grid. Each small square: x - axis, each unit is 1 minute. y - axis, each unit is 30 liters? Wait, no, the y - axis labels are 0,30,60,90,120,150,180,210,240,270,300,... So each grid line on y is 30 liters. x - axis: each grid line is 1 minute. So at x = 0, V = 240. At x = 6, let's see the line. From (0,240), moving 6 units to the right (x = 6) and how much up? Let's see the slope. Let's take another point, say x = 10, V = 360? Wait, 240 + 120 = 360, over 10 minutes? No, that would be slope 12. Wait, maybe my initial point is wrong. Wait, the graph is labeled "Amount of water in the tank (liters)" on the y - axis (vertical) and "Time (minutes)" on the x - axis (horizontal). Wait, no, actually, the vertical axis is V(x) (volume) and horizontal is x (time). So the graph is a line with x (time) on the horizontal and V(x) (volume) on the vertical. So when x = 0 (time = 0), V(0)=240. Then, let's find the equation of the line. Let's take two points: (0, 240) and (6,?). Wait, looking at the graph, when x = 6, let's see the y - value. Let's count the grid. From x = 0 (240) to x = 6, how much does it go up? Let's see, the line passes through (0, 240) and let's say (6, 240+620)? Wait, maybe the slope is 20 liters per minute. So \( V(x)=240 + 20x \). Then, when x = 6, \( V(6)=240+206=240 + 120 = 360 \). Wait, that makes sense. Let's check with another point. If x = 10, \( V(10)=240 + 2010 = 440 \)? Wait, no, maybe my slope is wrong. Wait, maybe the graph is from (0, 240) to (10, 360)? Then slope is \( \frac{360 - 240}{10 - 0}=\frac{120}{10}=12 \). Then \( V(x)=240 + 12x \). Then at x = 6, \( V(6)=240+126=240 + 72 = 312 \)? No, that doesn't match. Wait, maybe I misread the graph. Wait, the y - axis has labels 0,30,60,90,120,150,180,210,240,270,300,330,360,390,420,... So each major grid line is 30 liters. The x - axis: 0,1,2,3,4,5,6,7,8,9,10,... minutes. The line starts at (0, 240) ([SSE onError error]