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 is shown for various times (in minutes) in the following table.
Time(min) Temperature (F)
15 2200
20 2150
30 2050
60 1750
a.) Find a linear model for the temperature of the pottery after t minutes
b.) Explain the meaning of the slope of this line in the context of the problem.
c.) Assuming that the temperature continues to decrease at the same rate, what will be the temperature of the pottery in 3 hours?
A.) To find a linear model for the temperature of the pottery after t minutes, we need to find the equation of a straight line that best fits the given data points. We can use the two-point form of a linear equation to determine the slope and y-intercept.
Let's choose two points from the table, such as (15, 2200) and (60, 1750).
Using the two-point form formula, the equation of the line can be written as:
(y - y1) = ((y2 - y1) / (x2 - x1))(x - x1)
Substituting the given values:
(y - 2200) = ((1750 - 2200) / (60 - 15))(x - 15)
Simplifying further:
(y - 2200) = (-450 / 45)(x - 15)
(y - 2200) = -10(x - 15)
Expanding:
y - 2200 = -10x + 150
Rearrange to get the linear model equation:
y = -10x + 2350
Therefore, the linear model for the temperature of the pottery after t minutes is y = -10x + 2350.
B.) The slope of the line in the context of the problem represents the rate at which the temperature of the pottery is decreasing per minute. In this case, the slope is -10. This means that for every minute that passes, the temperature of the pottery is decreasing by 10 degrees Fahrenheit.
C.) To find the temperature of the pottery in 3 hours (which is 180 minutes), we can substitute t = 180 into the linear model equation:
y = -10(180) + 2350
Simplifying:
y = -1800 + 2350
y = 550
Therefore, the temperature of the pottery in 3 hours will be 550 degrees Fahrenheit.