You have already determined that the change in cups solds is 4 when the change in temperature is 2, so the slope of the function of cups versus temperature is 4/2 = 2. You are given three points (t, c), but you only need one of them to write the equation for cups in terms of temperature.
a) The equation of a line in point-slope form can be written as
... y = m(x - h) + k . . . . . . for some slope m and some point (h, k), where x is the input and y is the output.
You have an output of c, an input of t, a slope of 2, and a point (t, c) = (65, 20). Substituting these values into the above equation gives
... c = 2(t -65) +20
... c = 2t -110
b) If the pattern continues, all we need to do is use t=85 in the equation to find the number of cups sold.
... c = 2·85 -110 = 60
c) There are several limitations to "realistic" including the maximum expected temperature in the area, the rate at which lemonade can be produced and money taken during the hours of operation, available resources for producing lemonade, and probably additional considerations.
For the purpose here, we will assume the pattern continues and the limitation is the maximum temperature--perhaps 105 degrees. Then the maximum number of cups sold would be
... c = 2·105 -110 = 100
d) The slope was found at the beginning to be 2. This means the number of cups sold goes up by 2 when the maximum daily temperature goes up by 1 degree.
e) The c-intercept is -110. It represents the predicted number of cups sold when the temperature is 0 degrees. Maggie cannot sell less than 0 cups, a value predicted to be seen when the maximum temperature is 55 degrees.
