The Problem

The linear function given by models the number of radio stations on the air from 1950 to 1995. (Source: National Association of Broadcasters.)  Determine an equation whose solution is the year when there were 10000 radio stations on the air and then solve the equation graphically, symbolically, and numerically.

The number 10000 in this problem represents the value of  . So replacing with 10000 gives us the following equation to solve:

10000 = 214.2 (x – 1950) + 2322


Solving the Equation Numerically

To solve the equation numerically we use the TABLE feature on the calculator.  First, the equation to be solved must be entered in as in Intersection of Graphs Method.  Press  and then enter 10000 (the left side of the equation) for and enter 214.2 (X – 1950) + 2322 (the right side of the equation) for.  On your calculator you should see:

Screen Picture of TI-83 Calculator

 

To create a table press 2nd then WINDOW [TBLSET] to set up the necessary parameters. Enter 1980 for TblStart, and enter 1 for .  Make sure that both boxes are over the word AUTO and not over the word ASK.  If the box is on ASK, arrow down to the line and press ENTER as the box is blinking on AUTO.

Screen Picture of TI-83 Calculator

Now press 2nd  and then GRAPH [TABLE] and the table will appear.  Looking in the column, find the value that is closest to 10000.  This value is 10033 which corresponds to the year X = 1986.0.   Thus X 1986 since this is where and are closest to being equal.   (See Using a Table to Solve Equations.)

Screen Picture of TI-83 Calculator Screen Picture of TI-83 Calculator

 

In order to be more precise, once again press 2nd and then WINDOW [TBLSET].  Enter 1985 for TblStart and 0.1 for . We choose 1985 for a starting point since the solution appears to be after 1985 but before 1986. Then press 2nd GRAPH and another table will appear on your screen.

Screen Picture of TI-83 Calculator Screen Picture of TI-83 Calculator

Use the down arrow to scroll down the screen looking for the value in that is closest to 10000. This value is 9990.4 which corresponds to X = 1985.8. Thus, the refined solution is approximately 1985.8.  This result is consistent with our other two methods. 

Screen Picture of TI-83 Calculator


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