A scientist has a container of pure water. She raises the temperature of the water to its boiling point, then measures the mass of the water. The scientist adds a constant amount of energy to the container so the water will evaporate, and she records the remaining mass of the water every 10 minutes.
Time (minutes) Mass of Water (grams)
0 900
10 822
20 754
30 670
40 597
50 525
A. What is the domain of the function representing the data in the table?
Constructed Response Question
B. Calculate the average rate of change of the mass of the water between 0 and 50 minutes. Show your work and explain what this rate means in this situation.
Constructed Response Question
C. How long will it take for all of the water to evaporate? Show your work or explain your reasoning.
A. The domain of the function representing the data in the table is the set of all possible input values, which in this case is the set of time values recorded in the table. Therefore, the domain is {0, 10, 20, 30, 40, 50}.
B. To calculate the average rate of change of the mass of the water between 0 and 50 minutes, we need to find the change in mass and divide it by the change in time.
Change in mass = final mass - initial mass = 525 - 900 = -375 grams
Change in time = final time - initial time = 50 - 0 = 50 minutes
Average rate of change = change in mass / change in time = -375 / 50 = -7.5 grams per minute
The negative sign indicates a decrease in mass over time. In this situation, the average rate of change of the mass of the water represents the rate at which the water is evaporating. Therefore, the average rate of change of -7.5 grams per minute means that the water is evaporating at a rate of 7.5 grams per minute.
C. To determine how long it will take for all of the water to evaporate, we need to find the time when the mass of the water reaches 0 grams. From the given data, the mass of the water decreases by approximately 7.5 grams every minute.
Since the initial mass of the water is 900 grams and the rate of change is -7.5 grams per minute, we can set up an equation to represent the relationship between time (t) and mass (m):
m = 900 - 7.5t
To find the time when the mass reaches 0 grams, we set the equation equal to 0 and solve for t:
0 = 900 - 7.5t
7.5t = 900
t = 900 / 7.5
t = 120 minutes
Therefore, it will take 120 minutes for all of the water to evaporate.
A. The domain of the function representing the data in the table is the set of all possible input values, which in this case is the set of time values from 0 to 50 minutes. So the domain is [0, 50].
B. To calculate the average rate of change of the mass of the water between 0 and 50 minutes, we can use the formula:
Average rate of change = (change in mass) / (change in time)
We can calculate the change in mass by subtracting the initial mass (at 0 minutes) from the final mass (at 50 minutes):
Change in mass = 525 - 900 = -375 grams
The change in time is 50 - 0 = 50 minutes.
Average rate of change = (-375 g) / (50 min) = -7.5 g/min
The negative sign indicates that the mass of the water is decreasing over time. In this situation, the average rate of change of -7.5 g/min means that on average, the mass of the water is decreasing by 7.5 grams every minute.
C. To determine how long it will take for all of the water to evaporate, we can analyze the trend in the data. From the table, it can be observed that the mass of the water steadily decreases over time. Assuming this trend continues, at some point the mass of the water will reach 0, indicating that all of it has evaporated.
Based on the given data, it takes 50 minutes for the mass of the water to decrease from 900 grams to 525 grams. If the decrease in mass continues at a constant rate, it can be estimated that the water will completely evaporate after another 50 minutes, making a total of 100 minutes.
However, it is important to note that this estimation assumes that the rate of evaporation remains constant throughout the process. In reality, there might be factors that could affect the rate of evaporation, so the actual time required for all of the water to evaporate could be different.
A. In this context, the domain of the function representing the data in the table is the set of all possible values for the independent variable, which is time. From the given table, we can see that the time values are 0, 10, 20, 30, 40, and 50 minutes. Therefore, the domain of the function is {0, 10, 20, 30, 40, 50}.
B. To calculate the average rate of change of the mass of the water between 0 and 50 minutes, we need to determine the change in mass and divide it by the change in time.
The change in mass is found by subtracting the initial mass from the final mass:
Change in mass = Final mass - Initial mass
From the table, we can see that the initial mass at 0 minutes is 900 grams, and the final mass at 50 minutes is 525 grams:
Change in mass = 525 grams - 900 grams = -375 grams
The change in time is simply the difference between the final time and the initial time:
Change in time = Final time - Initial time = 50 minutes - 0 minutes = 50 minutes
Now we can calculate the average rate of change:
Average rate of change = Change in mass / Change in time
Average rate of change = (-375 grams) / (50 minutes)
Average rate of change = -7.5 grams/minute
In this situation, the average rate of change of the mass of water between 0 and 50 minutes is -7.5 grams per minute. This means that, on average, the mass of the water decreases by 7.5 grams every minute during this time interval.
C. To determine how long it will take for all of the water to evaporate, we can look at the trend in the data. As time progresses, the mass of water is decreasing. Eventually, the mass will reach zero when all the water has evaporated.
Looking at the table, we see that the mass decreases from 900 grams at 0 minutes to 525 grams at 50 minutes. The difference is 375 grams.
If the water is evaporating at a constant rate, we can estimate the time it will take for all the water to evaporate by assuming a linear relationship between time and mass. We can use the average rate of change we calculated earlier (-7.5 grams/minute) to estimate how long it will take for the remaining 525 grams to reach zero.
If we divide the remaining mass of 525 grams by the average rate of change of -7.5 grams/minute, we get:
Time = Remaining mass / Average rate of change
Time = 525 grams / (-7.5 grams/minute)
Time = -70 minutes
However, it is important to note that negative time does not make sense in this context. Therefore, we should exclude the negative solution. Thus, the estimated time it will take for all the water to evaporate is 70 minutes.