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) =
(b) Assuming the temperature continues to decrease at the same rate, what will be the temperature of the pottery in 3 hours? ________
since the relation is supposed to be linear, let's pick an "easy" pair, (20,2100) and (30,2000)
slope= (2100 - 2000)/(20-30)
= 100/-10 = - 10
so temp = -10t + b , where t is time in minutes
2000 = -10(30) + b
b = 2300
temp - =10t + 2300
b) for 3 hrs, temp = -10(180) + 2300
= 500
To find a linear model for the temperature of the pottery, we need to find the equation of a straight line that best represents the relationship between time and temperature.
(a) Let's use the equation of a straight line, y = mx + b, where y represents the temperature and x represents time. We need to find the slope (m) and the y-intercept (b).
To find the slope, we can use the formula:
m = (change in y) / (change in x) = (temperature at a later time - temperature at an earlier time) / (time at a later time - time at an earlier time)
For example, using the first two data points (15 min, 2150°F) and (20 min, 2100°F):
m = (2100 - 2150) / (20 - 15) = -10°F/min
Now that we have the slope, we can find the y-intercept (b) by substituting one of the data points into the equation:
Using the first data point (15 min, 2150°F):
2150 = (-10)(15) + b
2150 = -150 + b
b = 2300°F
Therefore, the linear model for the temperature of the pottery after minutes is:
T(t) = -10t + 2300
(b) We need to convert 3 hours to minutes. Since there are 60 minutes in an hour, 3 hours is equal to 3 x 60 = 180 minutes.
To find the temperature of the pottery after 3 hours (or 180 minutes), we can plug 180 into our linear model:
T(180) = -10(180) + 2300
T(180) = -1800 + 2300
T(180) = 500°F
Therefore, the temperature of the pottery after 3 hours (or 180 minutes) is 500°F.