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
(a) 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 (40, 760), (60, 1600), (70, 2020), and (85, 2650).
First, we need to find the slope of the line. The slope formula is given by:
slope = (change in y)/(change in x)
Using the points (40, 760) and (60, 1600):
slope = (1600 - 760)/(60 - 40) = 840/20 = 42
So the slope of the line is 42.
Next, we need to find the y-intercept. We can choose any point to substitute into the equation of a line. Let's use (40, 760):
760 = 40*42 + 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
(b) The meaning of the slope of this line in the context of this problem is that for every 1-degree increase in temperature, an additional 42 acre-feet of water will evaporate. Therefore, the correct statement is:
The value of the slope means that an additional 42 af evaporate for a 1° increase in temperature.
(c) To find out how many acre-feet of water will evaporate per day when the temperature is 75°F, we can substitute T = 75 into the linear model:
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.
To find the linear model for the number of acre-feet of water that evaporate as a function of temperature, we need to find the equation of the line that represents the relationship between temperature and evaporation.
We can use the two points (60, 1600) and (70, 2020) from the table to determine the slope of the line.
Slope (m) = (change in evaporation)/(change in temperature)
= (2020 - 1600)/(70 - 60)
= 420/10
= 42
Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line.
Using the point (60, 1600) and the slope of 42, we have:
E - 1600 = 42(T - 60)
Simplifying the equation, we get:
E = 42T - 240
So, the linear model for the number of acre-feet of water that evaporate as a function of temperature is:
E(T) = 42T - 240
Now let's explain the meaning of the slope in the context of this problem. The slope of the line is 42, which means that an additional 42 acre-feet of water evaporate for a 1-degree increase in temperature. Therefore, the correct answer is:
The value of the slope means that an additional 42 af evaporate for a 1° increase in temperature.
To find how many acre-feet of water will evaporate per day when the temperature is 75°F, we can substitute T = 75 into the linear equation we found earlier:
E(75) = 42(75) - 240
= 3150 - 240
= 2910 acre-feet
Therefore, when the temperature is 75°F, approximately 2910 acre-feet of water will evaporate per day.