A trainer for a professional football team keeps track of the amount of water players consume throughout practice. The trainer observes that the amount of water consumed is a linear function of the temperature on a given day.
The trainer finds that when it is 90 degrees the players consume about 220 gallons of water, and when it is 76 degrees the players consume about 178 gallons of water.
Part A: Write a linear function to model the relationship between the gallons of water consumed and the temperature.
Part B: Explain the meaning of the slope in the context of the problem.
Part A: To write a linear function to model the relationship between the gallons of water consumed and the temperature, we can use the slope-intercept form of a linear equation, which is y = mx + b.
Let's assign the temperature as x and the gallons of water consumed as y. We are given two data points: (90, 220) and (76, 178).
Using the first data point, we can substitute x = 90 and y = 220 into the linear equation:
220 = 90m + b (Equation 1)
Using the second data point, we can substitute x = 76 and y = 178 into the linear equation:
178 = 76m + b (Equation 2)
Now, we have a system of two equations with two unknowns (m and b). We can solve this system of equations to find the values of m and b.
Subtracting Equation 2 from Equation 1, we eliminate b:
220 - 178 = 90m - 76m
42 = 14m
Dividing both sides by 14:
m = 42/14
m = 3
Substituting the value of m into Equation 1 or Equation 2:
220 = 90(3) + b
220 = 270 + b
b = -50
Therefore, the linear function that models the relationship between the gallons of water consumed (y) and the temperature (x) is:
y = 3x - 50
Part B: The slope (m = 3) in the context of the problem represents the rate of change in the gallons of water consumed per unit change in temperature. It tells us that for every 1-degree increase in temperature, the players consume 3 more gallons of water. Similarly, for every 1-degree decrease in temperature, the players consume 3 fewer gallons of water. In other words, the slope indicates the constant rate at which the consumption of water changes with respect to temperature.
Part A: To write a linear function to model the relationship between the gallons of water consumed and the temperature, we need to determine the slope and the y-intercept.
Let's use the information given:
When the temperature is 90 degrees, the players consume about 220 gallons of water, and when it is 76 degrees, the players consume about 178 gallons of water.
We can use these two points to find the equation of the line.
First, let's determine the slope:
Slope (m) = (change in y) / (change in x)
Change in y = 220 - 178 = 42 (gallons)
Change in x = 90 - 76 = 14 (degrees)
Slope (m) = 42 / 14 = 3 (gallons/degree)
Now, let's determine the y-intercept:
Using the point (90, 220), we can substitute the values into the equation y = mx + b and solve for b (the y-intercept).
220 = 3(90) + b
220 = 270 + b
b = -50
So the linear function that models the relationship between the gallons of water consumed (y) and the temperature (x) is:
y = 3x - 50.
Part B: The slope in this context represents the rate of change in the amount of water consumed per degree change in temperature.
In this case, the slope of 3 (gallons/degree) means that for every 1-degree increase in temperature, the players consume an additional 3 gallons of water. Similarly, for every 1-degree decrease in temperature, the players would consume 3 gallons less water.
Thus, the slope reflects the relationship between temperature and water consumption, indicating how much the water consumption changes with each degree change in temperature.