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?
You are given the XY coordinates. Take Time to be X, and Temperature to be Y. Create a graph on a Cartesian coordinate system.
A) If done correctly your linear Regression should be Y=-10X + 2350.
B) Your slope is -10. Meaning that for every 1 minute the pottery cools by 10 degrees.
C) From B we can see that the temperature will continue to cool 10 degrees every minute. There are 60 minutes in an hour X two hours gives us 120 minutes. Multiply that 120 minutes by -10 degrees per minute. And you get -1200 degrees. You then add this to your final temperature at the one hour mark to get a final temperature at three hours as 550 degrees F.
a.) To find a linear model for the temperature of the pottery after t minutes, we can use the slope-intercept form of a linear equation, y = mx + b, where y is the temperature and x is the time in minutes.
Let's start by finding the slope, m, which represents how much the temperature changes per minute.
m = (change in temperature) / (change in time)
m = (1750 - 2200) / (60 - 15)
m = -450 / 45
m = -10
Now we can substitute the slope, m, and any point (x, y) on the line into the equation and solve for the y-intercept, b.
Using the point (15, 2200):
2200 = -10 * 15 + b
2200 = -150 + b
b = 2350
Thus, the linear model for the temperature of the pottery after t minutes is:
Temperature = -10t + 2350
b.) The slope of the line in the context of the problem represents the rate at which the temperature decreases per minute. In this case, the slope is -10, which means that the temperature decreases by 10 degrees Fahrenheit every minute.
c.) To find the temperature of the pottery in 3 hours (which is 180 minutes), we can substitute t = 180 into the linear model we found in part a.
Temperature = -10 * 180 + 2350
Temperature = -1800 + 2350
Temperature = 550
Therefore, the temperature of the pottery is expected to be 550 degrees Fahrenheit after 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 line that best fits the given data points. We can use the formula for finding the equation of a line: 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.
Using the data points given, we can find the slope, m, first. Let's choose two points, (15, 2200) and (60, 1750), and use the slope formula: m = (y2 - y1) / (x2 - x1).
m = (1750 - 2200) / (60 - 15)
= -450 / 45
= -10
Now that we have the slope, we can find the y-intercept, b. We can choose any point on the line, let's use (15, 2200):
y = mx + b
2200 = -10 * 15 + b
2200 = -150 + b
b = 2200 + 150
b = 2350
Therefore, the linear model for the temperature of the pottery after t minutes is:
Temperature = -10t + 2350
b.) In the context of the problem, the slope of the line (-10, in this case) represents the rate at which the temperature of the pottery is decreasing per minute. For every minute that passes, the temperature decreases by 10 degrees Fahrenheit.
c.) To find the temperature of the pottery in 3 hours, we need to convert hours to minutes since our model is based on minutes. Since there are 60 minutes in an hour:
3 hours = 3 * 60 minutes = 180 minutes
Now we can substitute t = 180 minutes into our linear model:
Temperature = -10t + 2350
Temperature = -10 * 180 + 2350
Temperature = -1800 + 2350
Temperature = 550
Therefore, the temperature of the pottery in 3 hours would be 550 degrees Fahrenheit.