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?
note the slope from point to point:
(2150-2200)/(20-15) = -50/5 = -10
(2050-2150)/(30-20) = -100/10 = -10
(1750-2050)/(60-30) = -300/30 = -10
So, it looks like
F = -10T
But, F(15) = 2200, not -150
SO, making that adjustment, we have
F = 2350 - 10T
a.) To find a linear model for the temperature of the pottery after t minutes, we can use the data provided to determine the equation of a straight line that passes through these points.
One way to do this is by using the slope-intercept form of a linear equation, y = mx + b, where y is the dependent variable (temperature in this case), x is the independent variable (time in minutes), m is the slope, and b is the y-intercept.
To find the slope, m, we can use the formula:
m = (y2 - y1) / (x2 - x1)
Choosing two points from the table, (15, 2200) and (20, 2150), we can plug in the values:
m = (2150 - 2200) / (20 - 15)
= -10
Now that we have the slope, we need to find the y-intercept, b. We can do this by selecting any point from the table and substituting the values into the slope-intercept form:
Using point (15, 2200):
2200 = -10(15) + b
b = 2350
Therefore, the linear model for the temperature of the pottery after t minutes is:
Temperature = -10t + 2350
b.) The slope of this line, -10 in this case, represents the rate at which the temperature is decreasing over time. It tells us that for every increase of 1 minute, the temperature decreases by 10 degrees Fahrenheit.
c.) To find the temperature of the pottery in 3 hours, we need to convert the time from hours to minutes. Since there are 60 minutes in 1 hour, 3 hours would be 3 * 60 = 180 minutes.
Plugging this value into the linear model we found earlier:
Temperature = -10(180) + 2350
= -1800 + 2350
= 550
Therefore, the temperature of the pottery in 3 hours would be 550 degrees Fahrenheit.