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 use the given data points to find the equation of a line that relates the time (t) and the temperature (F). We can use the slope-intercept form of the equation of a line, which is y = mx + b, where y is the dependent variable (temperature in this case), x is the independent variable (time in this case), m is the slope of the line, and b is the y-intercept.
Let's start by finding the slope (m) of the line. We can choose any two data points to calculate the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Let's choose the points (15, 2200) and (60, 1750):
m = (1750 - 2200) / (60 - 15)
m = -450 / 45
m = -10
Now that we have the slope, we can find the y-intercept (b) using one of the data points. Let's use the point (15, 2200):
y = mx + b
2200 = -10(15) + b
2200 = -150 + b
b = 2350
So the linear model for the temperature of the pottery after t minutes is:
F = -10t + 2350
b.) The slope of the line (-10 in this case) represents the rate at which the temperature of the pottery decreases per minute. The negative sign indicates that the temperature is decreasing. In this context, the slope tells us that for every minute that passes, the temperature of the pottery decreases by 10 degrees Fahrenheit.
c.) To find the temperature of the pottery after 3 hours, we need to convert the 3 hours into minutes (since the data is given in minutes).
1 hour = 60 minutes
So, 3 hours = 3 x 60 = 180 minutes
We can then substitute this value into our linear model:
F = -10t + 2350
F = -10(180) + 2350
F = -1800 + 2350
F = 550
Therefore, the temperature of the pottery after 3 hours (180 minutes) will be 550 degrees Fahrenheit.