Associated Topics || Dr. Math Home || Search Dr. Math

### Using Graphs to Solve Problems

```
Date: 12/29/96 at 16:02:46
From: Jerry W. Gambill
Subject: Precalculus

Hi Dr. Math,

I am a firefighter in California and several of us are taking a
precalculus class and working on it through the mail.  We have
recently come upon two problems that none of us can seem to get on the
right track to solving.  Thought you might lend some support.

(1) A. A company has a fixed monthly operating cost of \$120,000 and
a variable cost per unit manufractured of \$50.00.  Draw the graph of
the monthly total cost vs. units manufractured.

B. The company sells the units at \$70.00 each but on the average
5 percent of the manufactured units are sold at cost, \$50.00.  Draw
the line of the company profits vs units manufactured (X1000).
Calculate how many units must be manufactured to cover the fixed
operating costs.

(2) A company profit for x units sold is defined by the equation
P = 100x - .1x^2.  Sketch P vs x and find the company's maximum
profit.  Indicate the range of x (units sold) for the company to have
a profit.

```

```
Date: 12/30/96 at 11:01:34
From: Doctor Rob
Subject: Re: Precalculus

We applaud your efforts to learn calculus!

In problem (1) A, take different values of X, such as 0, 200, 400,
and so on, and calculate the monthly total cost = monthly fixed cost
plus monthly variable cost (the latter is X*\$50.00).  Call these
numbers the Y-values.  Plot the points (X, Y) in the XY-plane, then
connect the dots.  Negative values of X should probably be ignored
since a negative total cost doesn't make much sense in this situation.

In (1) B, 95 percent of the X units manufactured are sold at \$70, and
5 percent of the X units manufactured are sold at \$50.  This leads to
a total income of 0.95 * X * \$70 + 0.05 * X * \$50 = Y.  For the same
values of X (0, 200, 400, etc.), compute the value of this new Y.
Plot the points (X, Y) in the XY-plane, and again connect the new
dots.

The question asks for the point where the two graphs cross.  This is
found by solving the two equations of the two graphs simultaneously,
to get the values of X and Y that make both equations true at the same
time.

In (2), again pick convenient values of X (0, 200, 400, etc.) and
compute the corresponding values of P = 100*X - 0.1*X^2.  Plot the
points (X,P) in the XY-plane, and connect them.  This time the graph
will not be a straight line, but a curve.  The question is asking for
the highest point on the curve (largest P value), and the range of
X-values for which the curve lies above the X-axis (positive P
values).

In all these sketching problems, if you think something interesting
or important is going on between the points you have chosen, pick more
values of X in between those points and plot the corresponding points.
This will give you a more detailed view of the graph in that region.

If you need more help, feel free to write to us again.

-Doctor Rob,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search