The Snowtree cricket behaves in a rather interesting way: The rate at which it chirps depends linearly on the temperature. One summer evening you hear a cricket chirping at a rate of 160 chirps/minute, and you notice that the temperature is 80°F. Later in the evening, the cricket has slowed to 120 chirps/minute, and you notice that the temperature has dropped to 70°F. Express the temperature T as a linear function of the cricket's rate of chirping r.
treat it as two ordered pairs of the type (t,c)
given: (80,160) and (70, 120)
slope = (160-120)/(80-70) = 40/10 = 4
so
chirps = 4xtime + b
using (80,160)
160 = 4(80) + b
b = -160
chirps = 4(time) - 160
or
4(time) = chirps + 160
time = (1/4)chirps + 40
testing: if chirps = 120
time = (1/4)(120) + 40
= 30 + 40 = 70 , as given
Hello Philo !Je ne sais pas d'où viennent mes cercles à pâtisserie : mon frère me les a offerts pour mes 30 ans.En revanche, je sais où tu peux en trouver : chez Dehillerin, à Paris, pas très loin de Chttnele¢.SinoÃ, j'imagine que sur meilleurduchef il doivent en vendre.A bientôt !
To express the temperature T as a linear function of the cricket's rate of chirping r, we can use the concept of linear regression.
Linear regression finds the equation of a line that best fits the given data points. In this case, the data points are the temperature (T) and the rate of chirping (r).
We are given two data points:
Point 1: (Temperature 80°F, Chirps/minute 160)
Point 2: (Temperature 70°F, Chirps/minute 120)
To find the equation of the line, we need to calculate the slope and the y-intercept.
Step 1: Calculate the slope (m):
The slope of a line is given by the formula:
m = (change in y) / (change in x)
m = (Chirps/minute 2 - Chirps/minute 1) / (Temperature 2 - Temperature 1)
m = (120 - 160) / (70 - 80)
m = -40 / -10
m = 4
Step 2: Calculate the y-intercept (b):
The y-intercept of a line is given by the formula:
b = y - mx
Taking the first point (80°F, 160 chirps/minute):
160 = 80 * 4 + b
160 = 320 + b
b = 160 - 320
b = -160
Therefore, the equation of the line is:
T = 4r - 160
So, the temperature T can be expressed as a linear function of the cricket's rate of chirping r using the equation T = 4r - 160.
To express the temperature T as a linear function of the cricket's rate of chirping r, we need to determine the equation of the linear relationship between temperature and chirping rate.
We are given two sets of data points: when the chirping rate is 160 chirps/minute, the temperature is 80°F, and when the chirping rate is 120 chirps/minute, the temperature is 70°F.
We can use the formula for a linear relationship between two variables: y = mx + b, where y represents the dependent variable (temperature), x represents the independent variable (chirping rate), m represents the slope of the line, and b represents the y-intercept.
Let's solve for the slope (m) first. The slope of a line can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Using the coordinates (160 chirps/minute, 80°F) and (120 chirps/minute, 70°F):
m = (70 - 80) / (120 - 160)
= -10 / (-40)
= 1/4
So the slope (m) is 1/4.
Now let's find the y-intercept (b) using the point-slope form of a linear equation:
y - y1 = m(x - x1)
Using the coordinates (160 chirps/minute, 80°F):
y - 80 = (1/4)(x - 160)
y - 80 = (1/4)x - 40
y = (1/4)x + 40
Therefore, the linear function expressing the temperature T as a function of the cricket's rate of chirping r is:
T = (1/4)r + 40