A fish tank is being filled with water. The graph below shows the relationship between the amount of water W (in liters) in the tank and the time T (in minutes) since the tank started being filled.
Amount of water (liters)
W20406080100120140160180200T1234567890
Time (minutes)
(a)Write an equation to represent the relationship between W and T.
(b)Choose the correct statement to represent this relationship.
The fish tank is being filled with 1 liter of water per minute.
The fish tank is being filled with 40 liters of water per minute.
The fish tank is being filled with 1 liter of water per 20 minutes.
The fish tank is being filled with 20 liters of water per minute.
(a) The equation to represent the relationship between W and T is W = 20T.
(b) The correct statement to represent this relationship is: The fish tank is being filled with 20 liters of water per minute.
(a) To write an equation representing the relationship between W (water in liters) and T (time in minutes), we can observe that the amount of water in the tank is increasing at a constant rate. This means that the tank is being filled with a constant amount of water every minute.
The equation can be written as:
W = mT + b
where m is the slope (rate of change) representing the amount of water being added per minute, and b is the y-intercept representing the initial amount of water in the tank.
From the graph, we can see that the tank starts with 20 liters of water (y-intercept) and that after 10 minutes, the tank has 40 liters of water. Therefore, we can calculate the slope (m) using the formula:
m = (change in water) / (change in time)
m = (40 - 20) / (10 - 0)
m = 20 / 10
m = 2
Plugging the slope and y-intercept into the general equation, we get:
W = 2T + 20
(b) The correct statement to represent this relationship is: The fish tank is being filled with 2 liters of water per minute.
In order to find the equation representing the relationship between the amount of water W and the time T, we need to analyze the graph. Based on the information provided in the graph, we can see that the amount of water in the tank increases linearly over time.
(a) To find the equation, we need to determine the slope of the line. The slope represents the rate at which the water level increases with respect to time. We can calculate the slope by finding the change in the amount of water (ΔW) divided by the change in time (ΔT) between two points on the line.
Let's choose two points on the line, for example, (0, 0) and (10, 100). The change in the amount of water is 100 - 0 = 100 liters, and the change in time is 10 - 0 = 10 minutes. Therefore, the slope is ΔW/ΔT = 100/10 = 10 liters per minute.
To find the equation, we can use the formula for a linear relationship, which is y = mx + b, where y is the dependent variable (W), x is the independent variable (T), m is the slope, and b is the y-intercept (the value of y when x = 0).
In this case, the equation would be W = 10T + b. To find the value of b, we can use one of the data points given in the graph. Let's use the point (0, 0), which means that when time T is 0, the amount of water W is 0. Plugging these values into the equation, we get 0 = 10(0) + b, which simplifies to 0 = b. Therefore, the equation representing the relationship between W and T is:
W = 10T
(b) Now that we have the equation W = 10T, we can determine which statement accurately represents this relationship.
The fish tank is being filled with 1 liter of water per minute: This statement is incorrect because the slope of the equation is 10, indicating that the tank is being filled with 10 liters of water per minute, not just 1.
The fish tank is being filled with 40 liters of water per minute: This statement is incorrect because the slope of the equation is 10, not 40.
The fish tank is being filled with 1 liter of water per 20 minutes: This statement is incorrect because the slope of the equation is 10, indicating that the tank is being filled with 10 liters of water per minute, not 1 liter per 20 minutes.
The fish tank is being filled with 20 liters of water per minute: This statement is incorrect because the slope of the equation is 10, not 20.
Therefore, the correct statement is: The fish tank is being filled with 10 liters of water per minute.