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.
We see that the temperature of the cake will get closer to room temperature as it cools but its temperature cannot be below room temperature. Hence, y=70 is the asymptote. However, the calculator exponential model assumes the asymptote is y=0. To accommodate it, we need to write another list by subtracting 70 from each temperature value.
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_3" , 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 transformed data.
Regression equation: y=127.27(0.84)^x
We need to translate it vertically by 70 units to model the original data. Therefore, the exponential model for the given data set is the following equation.
Exponential Model: y=127.27(0.84)^x+ 70
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.
b
To find how long it takes the cake to cool to 120^(∘)F, we will use the exponential model we found.
Exponential Model: y=127.27(0.84)^x+70We will find for which x-value, the equation is equal to 120. 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 120.
From the table, we see that when x=5 the function equals 123.23. To find a value close to 120, 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 120 when x=5.3. Therefore, the cake takes about 5.3 minutes to cool to 120^(∘)F.
