Monday, October 08, 2012

Algebra: Using Graphing Utilities to Solve Word Problems

Proper use of a graphing utility is the most important skill you should develop. Once you master the calculator (or other utility), you can perform any task and complete any problem. You then only need the mathematical skills necessary to set up the information for solving. You can avoid many tedious steps, solving problems the easy way, if you just know how to use your calculator.
In my College Algebra textbook, the very first section covers how to use your graphing utility to model word problems. We have already discussed the Calculate Menu and How to Input Functions—this post will cover how to interpret and solve word problems using your graphing utility of choice.
In this post, I’ll be using a program called Microsoft Mathematics for graph images, but will be instructing you with button and menu names from the TI-84 Plus Silver Edition. Any graphing calculator will perform these tasks, though you may need to consult your manual (or find it online) if you have a different calculator and cannot find the buttons or menus I mention here.
Let’s jump right in with our first word problem.
 First, I like to underline/circle/box (or highlight with various colors) pertinent information, to help me sort through the words and get straight to the facts. Underline the question or directive (red below). Circle plain information or direct substitutions (blue below). Box indirect substitutionary information; i.e. information that will help you create an equation (green below).
The second step is to draw a picture representing the problem, if possible.
Let’s consider the information we know. The blue, boxed information below is what we gather directly from the problem. Next, we must obtain equations. Since we are told the tank is a right circular cylinder, and since we are given a volume, it is reasonable to assume we need the formula for the volume of a right circular cylinder, which I’ve labeled “Equation A.” Consider the portion we boxed in green earlier—this sentence tells us a relationship between the radius and the height of the tank. With this information, we form “Equation B.”
We have a volume to substitute (given), but then we are left with Equation A containing two variables, r and h. Equation A and B form a system of equations with two variables. The best way to solve this is to rewrite Equation B in terms of one of the variables and then substitute that into Equation A and solve for the other variable. This method is called “Substitution.”
First, solve Equation B for h (you could solve for r and would obtain the same final answer). Let’s call this Equation C (purple box below).
Substitute Equation C for “h” in Equation A. Also, replace V with the volume given in the problem, 40,000 cubic feet. Simplify the right side and now we have Equation D.
This is the equation we need to solve. You’ll notice, however, that this is a cubic function that you cannot solve by hand (you could if the left side was zero). Therefore, we have to use a graphing utility to solve. There are a few ways of doing this.
1) Input each “side” of Equation D as a separate function in the Y= menu, then graph the functions and use the Table menu to locate their intersection points.
2) Input each “side” of Equation D as a separate function in the Y= menu, then use the Calculate menu to find their points of intersection.
3) Re-arrange Equation D so that one side is equal to zero, input the new function into the Y= menu and find the roots, or x-intercepts, of that equation.
Method (2) is the easiest, in my opinion, so that is what we’ll do. If you’ve seen my blog post on Graphing Functions, then you already know how to input each “side” of Equation D into your calculator as two separate functions. And if you’ve seen my blog post on the Calculate Menu, you already know how to find the intersection of two functions.
When you graph the two functions,
they should look something like this:
(Note: The blue horizontal line is Y1 and the green curve is Y2.)
Below, I have circled the intersection points we need to find in red.
Use the Calculate Menu (2nd, TRACE; 5: Intersect) to determine these points. If you need help, see my post on using the Calculate Menu here. Remember, these “x-values” are the radius values. Notice that a radius cannot be negative, so we can dismiss our first answer. We can find the corresponding height values by simply inserting each radius value into Equation B from earlier in the problem.
And that’s it! Many problems are solved using this method. Anytime you have an equation you cannot solve, you can input each side into the calculator as separate functions (if there is only one variable throughout) and then simply find the intersection points of the functions.
Another way you might use your calculator in Algebra is if you need to fill in a table of values. Instead of calculating each answer individually, you can input the equation as a function in your calculator. To find x-values that correspond to given y-values, use the first option in the Calculate Menu (1: value). To find y-values that correspond to given x-values, use the Table Menu (2nd, GRAPH).
I hope this post has helped you learn to use your calculator when solving algebra word problems.