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.

Thank you for your assistance.

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/