A piece of pottery is removed from a kiln and allowed to cool in a controlled environment. The temperature (in degrees Fahrenheit) of the pottery after it is removed from the kiln for various times (in minutes) is shown in the following table.
Time, min Temperature, °F
15 2150
20 2100
30 2000
60 1700
(a) Find a linear model for the temperature of the pottery after minutes.
T(t)=? _____
To find a linear model for the temperature of the pottery after minutes, we can use the formula for a line:
y = mx + b
Where:
y = temperature of the pottery
x = time in minutes
m = slope of the line
b = y-intercept
To find the values of m and b, we need two data points from the table. Let's select the points (15, 2150) and (60, 1700).
Using the formula for the slope (m) of a line:
m = (y2 - y1) / (x2 - x1)
m = (1700 - 2150) / (60 - 15)
m = -450 / 45
m = -10
Now, let's substitute the value of the slope (m) into the equation and use one of the data points to solve for the y-intercept (b).
Using the point (15, 2150):
2150 = -10(15) + b
2150 = -150 + b
b = 2300
So, the linear model for the temperature of the pottery after minutes is:
T(t) = -10t + 2300
To find a linear model for the temperature of the pottery after a certain number of minutes, we need to find the equation of a line that best fits the given data points.
We are given four data points: (15, 2150), (20, 2100), (30, 2000), (60, 1700).
The equation for a linear model is given by the equation: y = mx + b, where m is the slope of the line and b is the y-intercept.
To find the slope, m, we can use the formula:
m = (y2 - y1) / (x2 - x1)
Taking two of the data points, (15, 2150) and (60, 1700), we can substitute the values into the formula:
m = (1700 - 2150) / (60 - 15)
m = -450 / 45
m = -10
Now that we have the slope, we can find the y-intercept, b, by substituting the slope and one of the data points into the linear equation:
2150 = -10(15) + b
2150 = -150 + b
b = 2150 + 150
b = 2300
Therefore, the linear model for the temperature of the pottery after minutes is:
T(t) = -10t + 2300