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) =
(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?
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.
Let's use the two points (70, 2020) and (85, 2650).
First, calculate the slope of the line using the formula:
slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1)
slope = (2650 - 2020) / (85 - 70) = 630 / 15 = 42
So, the slope of the line is 42.
Next, we can use the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is one of the given points.
Let's use the point (70, 2020).
E(T) - 2020 = 42(T - 70)
Simplifying, we get:
E(T) = 42T - 2940 + 2020
E(T) = 42T - 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: an additional 42 acre-feet 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 simply substitute T = 75 into the linear model equation we found in part (a).
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.