Suppose a snowy-tree cricket chirps 180 times per minute if the temperature is 102 degrees, and chirps 120 times per minute when the temperature is 90 degrees. Construct a mathematical model that will predict the number of chirps per minute at a temperature in degrees Fahrenheit. (Write your answer as a function in terms of degrees Fahrenheit, expressing it in slope-intercept form.)
This is hardly calculus
y=mx+b
180=m102+b
120=m90+b
subtrract
60=12m solve for m, then b.
To construct a mathematical model that predicts the number of chirps per minute at a temperature in degrees Fahrenheit, we can use the concept of linear regression. Linear regression helps us find the equation of a straight line that best fits the given data points.
Let's define the temperature in degrees Fahrenheit as "x" and the number of chirps per minute as "y". We have two data points: (102, 180) and (90, 120).
To find the equation of the line in slope-intercept form (y = mx + b), we need to determine the slope (m) and the y-intercept (b).
First, we need to find the slope:
m = (y2 - y1) / (x2 - x1)
Using the coordinates of the two data points:
m = (120 - 180) / (90 - 102)
m = -60 / -12
m = 5
The slope of the line is 5.
Next, we can use the point-slope form of the equation to find the y-intercept:
y - y1 = m(x - x1)
Using the coordinates of one of the data points, let's take (102, 180):
y - 180 = 5(x - 102)
y - 180 = 5x - 510
y = 5x - 330
Therefore, the mathematical model that predicts the number of chirps per minute at a temperature in degrees Fahrenheit is:
chirps per minute = 5*temperature in degrees Fahrenheit - 330
In slope-intercept form, the equation is:
y = 5x - 330
The coefficient of x (5 in this case) represents the rate at which the number of chirps per minute changes with each degree Fahrenheit increase in temperature. The constant term (-330) represents the starting point or the number of chirps at 0 degrees Fahrenheit.