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 (in minutes) for which the mass of water is recorded. So, the domain of the function is {0, 10, 20, 30, 40, 50}.
B. To calculate the average rate of change of the mass of water between 0 and 50 minutes, we need to find the change in mass over the change in time.
Change in mass = Final mass - Initial mass = 525 - 900 = -375
Change in time = Final time - Initial time = 50 - 0 = 50
Average rate of change = Change in mass / Change in time = -375 / 50 = -7.5
The average rate of change of the mass of the water is -7.5 grams per minute. This means that for every minute that passes, on average, the mass of the water decreases by 7.5 grams.
C. To determine how long it will take for all of the water to evaporate, we can look for the point where the mass of water becomes zero. In the given data, the mass of water decreases by a constant rate of 7.5 grams per minute.
Using the average rate of change, we can set up the following equation:
900 - 7.5t = 0
Solving for t:
7.5t = 900
t = 900 / 7.5
t = 120
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 values for the time in minutes. In this case, the domain is {0, 10, 20, 30, 40, 50}, as these are the time points for which the mass of water is recorded.
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 over the change in time.
The change in mass is the difference between the initial mass (900 grams) and the final mass (525 grams) recorded in the table. It is calculated as:
Change in mass = Final mass - Initial mass
= 525 grams - 900 grams
= -375 grams
The change in time is the difference between the final time (50 minutes) and the initial time (0 minutes). It is 50 minutes - 0 minutes = 50 minutes.
Therefore, the average rate of change is:
Average rate of change = Change in mass / Change in time
= -375 grams / 50 minutes
= -7.5 grams per minute
In this situation, the average rate of change of -7.5 grams per minute means that, on average, the mass of 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 need to look for the time point at which the mass of water becomes 0 grams.
By examining the data in the table, we can see that the mass of water decreases by a constant amount every 10 minutes. Therefore, it would take 70 minutes (7 intervals of 10 minutes) for the mass of water to reach 0 grams.
Thus, all of the water will evaporate after 70 minutes.
A. To find the domain of the function representing the data in the table, we need to look at the values in the "Time (minutes)" column. The domain is the set of all possible inputs for the function. In this case, the time values in the table are: 0, 10, 20, 30, 40, and 50. 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 water between 0 and 50 minutes, we need to find the change in mass divided by the change in time.
First, we find the change in mass:
Mass at 50 minutes - Mass at 0 minutes = 525 - 900 = -375 grams
Next, we find the change in time:
50 minutes - 0 minutes = 50 minutes
Finally, we divide the change in mass by the change in time:
Average rate of change = (Change in mass)/(Change in time) = -375/50 = -7.5 grams/minute
The average rate of change of the mass of water between 0 and 50 minutes is -7.5 grams/minute. This means that, on average, the mass of the water is decreasing 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 need to observe the pattern in the data. As the time increases, the mass of the water decreases. At 0 minutes, the mass is 900 grams, and at 50 minutes, the mass is 525 grams.
Based on the given data, it seems that the mass of the water decreases by a constant amount every 10 minutes. Thus, we can assume that the rate of change observed from 0 to 50 minutes will continue to be constant.
If we take the average rate of change we calculated earlier (-7.5 grams/minute), we can use it to estimate the amount of time it will take for all of the water to evaporate.
Starting from a mass of 900 grams, we can calculate how many minutes it will take for the mass to reach 0 grams by dividing the initial mass by the average rate of change:
Time = Initial mass / Average rate of change = 900 grams / 7.5 grams/minute = 120 minutes
Therefore, it will take approximately 120 minutes for all of the water to evaporate.