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
Find a linear model for the number of acre-feet of water that evaporate as a function of temperature.
E(T) = _____
To find a 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 best fits the given data points.
We can use the slope-intercept form of a linear equation, which is y = mx + b, where y is the dependent variable, x is the independent variable, m is the slope, and b is the y-intercept.
In this case, the dependent variable is the number of acre-feet of water evaporated (E), and the independent variable is the temperature (T).
We can use two points from the given data to calculate the slope (m) of the line. Let's choose the points (40, 760) and (85, 2650).
The slope (m) can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (40, 760) and (85, 2650):
m = (2650 - 760) / (85 - 40)
m = 1890 / 45
m = 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. Let's choose the point (40, 760) as the point on the line.
Using the point-slope form: y - y1 = m(x - x1)
y - 760 = 42(x - 40)
Simplifying the equation:
y - 760 = 42x - 1680
y = 42x - 920
So the linear model for the number of acre-feet of water that evaporate as a function of temperature is:
E(T) = 42T - 920
To find a linear model for the number of acre-feet of water that evaporate as a function of temperature, we need to find the equation of a line that represents the relationship between the temperature (T) and the number of acre-feet of water evaporated (E).
We can use the formula for the equation of a line:
y = mx + b
where y is the dependent variable, x is the independent variable, m is the slope of the line, and b is the y-intercept.
From the given data, we can see that the temperature (T) is the independent variable and the number of acre-feet of water evaporated (E) is the dependent variable. We can assign T to x and E to y, so the equation becomes:
E = mx + b
To find the slope (m), we can use the formula:
m = (y2 - y1) / (x2 - x1)
Let's choose two data points from the table, for example, (40,760) and (60,1600):
m = (1600 - 760) / (60 - 40)
m = 840 / 20
m = 42
Now that we have the slope (m), we can find the y-intercept (b) by substituting the values of any data point into the equation and solving for b.
Let's choose the data point (40,760):
760 = 42(40) + b
760 = 1680 + b
b = 760 - 1680
b = -920
So the linear model for the number of acre-feet of water that evaporates as a function of temperature is:
E(T) = 42T - 920