Susan plants a tree in her backyard, and then she tracks the growth of the tree as shown in the chart below. Model this data with a logarithmic regression function, and use your model to answer the following questions.

  1. How tall will the tree be after 10 years?
  2. How many years will it take for the tree to grow to a height of 30 ft?
Year
Height (ft)
1
3
2
6
3
10
4
13
5
15
6
16

Note: To find a regression equation only, omit steps 1, 2, 3, and 7.

Step 1: Turn on the STAT PLOT; press 2nd, Y=, ENTER, ENTER.

Step 2: Clear the functions; press Y= then, CLEAR for each function that must be deleted.

Step 3: Select values for the viewing window; press WINDOW. Use the keypad to type the values from the table. Use the up and down arrow keys to scroll through the list.

Step 4: Go to the lists; press STAT, ENTER.

Step 5: Clear the lists; press the up arrow until the column heading is highlighted then, CLEAR, ENTER. Repeat Step 5 for all columns that have unneeded data. Use the left and right arrow keys to move between columns.

Step 6: Enter the data.

Step 7: View the data; press GRAPH.

Step 8: Choose the regression model; press STAT, right arrow.

Step 9: Select the logarithmic regression; press 9.

Step 10: Place the logarithmic regression into the function area; press VARS, right arrow, ENTER, ENTER, ENTER.

Step 11: View the scatter plot and regression model; press GRAPH.

The logarithmic regression function is y = 2.062 + 7.695 ln x.

  1. To predict how tall the tree will be after 10 years, means to find the value of y when x = 10.

Step 1: Go to the table of values; press 2nd, GRAPH. Use the down arrow to scroll to the required value of x = 10.

The height of the tree after 10 years is 19.78 ft.

One limitation of using the table of values to find the solution is that each column in the table is limited to six digits. It is important to be aware of this because, if your value has more than six digits, the digits at the end of the number will not be shown on the screen.

  1. To find the number of years needed for the tree to reach a height of 30 ft means find x when y = 30.

If you are asked to find this answer to the nearest year, you can use the table of values.

Step 1: Go to the table of values; press 2nd, GRAPH. Use the down arrow to scroll to the required value of y = 30.

At the beginning of year 37, the tree is not yet 30 ft tall. Therefore, the answer, to the nearest year, is 38.

If you are asked for a more precise answer, use the intersection feature that you were introduced to in Units 5 and 6.

Step 1: Input the given y-value; press Y=, down arrow to scroll to Y2, type in 30.

Step 2: Graph the functions; press GRAPH.

Step 3: Go to the CALCULATE menu; press 2nd, TRACE.

Step 4: Find the intersection of Y1 and Y2 ; press 5, ENTER, ENTER, ENTER.

The tree will be 30 ft tall after approximately 37.7 years.