Use a calculator to find an exponential model for the data set.
y=2144.23(0.95)^x
Practice makes perfect
We will first use a graphing calculator to find an exponential model for the given data set and then find how long it takes the glass to cool to 1200^(∘)F.
Writing an Exponential Model
We will use a graphing calculator to find an exponential model for the given data set. Press STAT, and choose Edit. Enter the values of time into the first list, and the values of temperature into the second list.
Since the room temperature is not given in the question, we assume that the temperature of the glass will get closer to zero. Hence, y=0 is the asymptote.
Now, press STAT again and navigate to the "CALC" menu. Move down to the ExpReg option and press ENTER once to select it. Follow the menu to select the lists L_1 and L_2, and once it is done move to the last line. Press ENTER to ask the calculator to calculate the regression.
The calculator shows you an exponential model for the given data.
Therefore, the exponential model for the given data set is the following equation.
Exponential Model: y=2144.23(0.95)^x
Checking Our Answer
Check Your Answer
To check the answer, we can draw the scatter plot and the regression line on the same graph. First, press WINDOW to set the viewing window.
With the values below, you will see the full scatter plot.
Next, you need to turn on the statistical plot.
Press 2ND Y= to bring up the plot selection menu. In the menu move to select one of the statistical plots and press ENTER.
Once the plot is selected choose the scatter plot type, and specify the lists where the data is stored.
Press Y= to check that both the scatter plot and the regression equation will be drawn. To see the graph, press GRAPH.
We see that the regression line fits the given data.
Using Model to Find Time
To find how long it takes the glass to cool to 1200^(∘)F, we will use the exponential model we found.
Exponential Model: y=2144.23(0.95)^x
We will find for which x-value the equation is equal to 1200. To do so, we will use the table features in a graphing calculator. To enter the equation on your calculator, push the button Y= and write the function.
By pushing 2nd and then GRAPH, we get a table of values for the whole number values of x. We want to find the x-value that makes the Y_1-column equal to 1200. From the table in the question, we expect the x-value to be greater than 10.
From the table, we see that when x=11 the function equal 1219.5. To find a value close to 1200, we need a table of x-values with one decimal digit. By pushing 2nd and WINDOW, we bring up table setup. Then, push 2nd and GRAPH once again to get table of values.
We see that Y_1 is approximately 1200 when x=11.3. Therefore, it takes about 11.3 minutes for the piece of glass to cool to 1200^(∘)F.
