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/
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