The rate at which water evaporates from a certain reservoir depends on the air temperature. The table below shows the number of acre-feet (af) of water per day that evaporate from the reservoir for various temperatures in degrees Fahrenheit.
Temperature, °F af
40 760
60 1600
70 2020
85 2650
(a) Find a linear model for the number of acre-feet of water that evaporate as a function of temperature.
E(T) = 1
(b) Explain the meaning of the slope of this line in the context of this problem.
The value of the slope means that an additional 1 af evaporate for a 42° increase in temperature.
The value of the slope means that an additional 42 af evaporate for a 1° increase in temperature.
The value of the slope means that an additional 1 af evaporate for a 42° decrease in temperature.
The value of the slope means that an additional 42 af evaporate for a 1° decrease in temperature.
(c) Assuming that water continues to evaporate at the same rate, how many acre-feet of water will evaporate per day when the temperature is 75°F?
3 af
Look at the data, using just the changes
(1600-760)af/(60-40)°F = 42 af/°F
(2020-1600)af/(70-60)°F = 42 af/°F
(2650-2020)af/(85-15)°F = 42 af/°F
I guess that tells the story, eh?
(a) The linear model for the number of acre-feet of water that evaporate as a function of temperature is E(T) = 42T - 2640.
(b) The meaning of the slope of this line in the context of this problem is: The value of the slope means that an additional 42 af evaporate for a 1° increase in temperature.
(c) Assuming that water continues to evaporate at the same rate, when the temperature is 75°F, the number of acre-feet of water that will evaporate per day is E(75) = 42(75) - 2640 = 975 af.
To find a linear model for the number of acre-feet of water that evaporate as a function of temperature, we can use the given data points to find the equation of a line that best fits the data.
We can use the formula for the equation of a line, y = mx + b, where y represents the number of acre-feet of water that evaporate and x represents the temperature.
Using the data points (40, 760) and (85, 2650), we can find the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
m = (2650 - 760) / (85 - 40)
= 1890 / 45
= 42
Using the point-slope form of a linear equation, we can substitute the slope (m) and one of the data points (40, 760) into the equation to find the y-intercept (b).
y = mx + b
760 = 42(40) + b
760 = 1680 + b
b = -920
Therefore, the linear model for the number of acre-feet of water that evaporate as a function of temperature is:
E(T) = 42T - 920
Now, let's interpret the slope (42) in the context of this problem.
(b) The value of the slope means that an additional 42 af evaporate for a 1° increase in temperature.
Now, to find the number of acre-feet of water that will evaporate per day when the temperature is 75°F, we can substitute T = 75 into the linear model we found in part (a):
E(75) = 42(75) - 920
= 3150 - 920
= 2230
Therefore, when the temperature is 75°F, approximately 2230 acre-feet of water will evaporate per day.
To find a linear model for the number of acre-feet of water that evaporate as a function of temperature, we can use the data given in the table.
Step 1: Choose two data points from the table. Let's choose (40, 760) and (85, 2650).
Step 2: Use the slope-intercept form of a linear equation (y = mx + b), where y represents the number of acre-feet of water evaporated and x represents the temperature in degrees Fahrenheit. We need to find the slope (m) and the y-intercept (b).
Step 3: Calculate the slope (m) using the formula:
m = (change in y) / (change in x)
m = (2650 - 760) / (85 - 40)
m = 1890 / 45
m = 42
So, the slope (m) of the linear model is 42.
Step 4: Plug in one of the data points into the linear equation to find the y-intercept (b). Let's use the point (40, 760).
760 = 42(40) + b
760 = 1680 + b
b = -920
So, the y-intercept (b) of the linear model is -920.
Therefore, the linear model for the number of acre-feet of water (E) that evaporate as a function of temperature (T) is:
E(T) = 42T - 920
(b) The slope of the line represents the rate of change in the number of acre-feet of water evaporated per degree Fahrenheit increase in temperature. In this context, the slope of 42 means that for every 1-degree increase in temperature, an additional 42 acre-feet of water will evaporate.
So, the correct answer is:
The value of the slope means that an additional 42 af evaporate for a 1° increase in temperature.
(c) To find the number of acre-feet of water that will evaporate per day when the temperature is 75°F, we can plug in T = 75 into the linear model E(T) = 42T - 920:
E(75) = 42(75) - 920
E(75) = 3150 - 920
E(75) = 2230
Therefore, when the temperature is 75°F, approximately 2230 acre-feet of water will evaporate per day.
So, the correct answer is:
Approximately 2230 acre-feet of water will evaporate per day when the temperature is 75°F.