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) = ? ____
(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?
_____ af

check the related questions below

To find a linear model for the number of acre-feet of water that evaporate as a function of temperature, we can use the equation of a linear relation:

y = mx + b

where y is the number of acre-feet of water evaporated (af), x is the temperature in degrees Fahrenheit (°F), m is the slope of the line, and b is the y-intercept.

Using the data given in the table:

Temperature, °F | af
40 | 760
60 | 1600
70 | 2020
85 | 2650

We can see that when the temperature is 40°F, the number of acre-feet evaporated is 760 (af). This gives us the point (40, 760).

Similarly, when the temperature is 85°F, the number of acre-feet evaporated is 2650 (af). This gives us the point (85, 2650).

Using these two points, we can find the slope of the line (m) using the formula:

m = (y2 - y1) / (x2 - x1)

m = (2650 - 760) / (85 - 40)
m = 1890 / 45
m = 42

So the slope of the line is 42.

To find the y-intercept (b), we can substitute the values of one of the points (40, 760) into the equation:

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 given by:

E(T) = 42T - 920

To find the number of acre-feet of water that 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 a linear model for the number of acre-feet of water that evaporate as a function of temperature, we need to determine the equation for a straight line that best fits the given data points.

We can use the slope-intercept form of a linear equation: y = mx + b, where y represents the dependent variable (number of acre-feet of water evaporated), x represents the independent variable (temperature), m represents the slope of the line, and b represents the y-intercept.

To calculate the slope (m), we use the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Using the points (40, 760) and (70, 2020), we have: m = (2020 - 760) / (70 - 40) = 1260 / 30 = 42.

Now, we can substitute the values of the slope (m) and one of the data points (e.g., (40, 760)) into the slope-intercept form to find the y-intercept (b).

Using the equation y = mx + b and substituting x = 40, y = 760, and m = 42, we have: 760 = 42 * 40 + b.

Simplifying the equation, we get: 760 = 1680 + b.

Now, we can solve for b: b = 760 - 1680 = -920.

Therefore, the linear model for the number of acre-feet of water evaporating as a function of temperature (E(T)) is given by: E(T) = 42T - 920, where T represents the temperature in degrees Fahrenheit.

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 equation.

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.