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 to find the equation of the line.
Let's define the temperature as "T" and the number of acre-feet of water that evaporate as "E(T)". We can use the formula for a linear equation:
E(T) = m * T + b
where "m" is the slope of the line and "b" is the y-intercept.
Using the given data points (40, 760), (60, 1600), (70, 2020), (85, 2650), we can calculate the slope "m".
m = (E2 - E1) / (T2 - T1)
= (1600 - 760) / (60 - 40)
= 840 / 20
= 42
Now we have the slope "m" as 42. To find the y-intercept "b", we can use any of the data points. Let's use (40, 760):
760 = 42 * 40 + b
760 = 1680 + b
b = -920
Therefore, our 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 is the rate at which the number of acre-feet of water evaporates per degree Fahrenheit increase in temperature. In this case, the slope is 42, so 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 how many acre-feet of water will evaporate per day when the temperature is 75°F, we can substitute T = 75 into our 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.