document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193131>Question 1193131</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For a certain company, the cost function for producing x items is C(x)=50x+250 and the revenue function for selling x items is R(x)=−0.5(x−130)2+8,450. The maximum capacity of the company is 170 items.<BR>\n" );
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document.write( "The profit function P(x) is the revenue function R(x) (how much it takes in)  minus the cost function C(x) (how much it spends). In economic models, one typically assumes that a company wants to maximize its profit, or at least make a profit!<BR>\n" );
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document.write( "Answers to some of the questions are given below so that you can check your work.<BR>\n" );
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document.write( "Assuming that the company sells all that it produces, what is the profit function?<BR>\n" );
document.write( "P(x)=   Preview Change entry mode .<BR>\n" );
document.write( "Hint: Profit = Revenue - Cost as we examined in Discussion 3.\r<BR>\n" );
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document.write( "What is the domain of P(x)?<BR>\n" );
document.write( "Hint: Does calculating P(x) make sense when x=−10 or x=1,000?\r<BR>\n" );
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document.write( "The company can choose to produce either 80 or 90 items. What is their profit for each case, and which level of production should they choose?\r<BR>\n" );
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document.write( "Profit when producing 80 items =     <BR>\n" );
document.write( "Profit when producing 90 items =    \r<BR>\n" );
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document.write( "Can you explain, from our model, why the company makes less profit when producing 10 more units?<BR>\n" );
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document.write( "Question 1) What is the profit function?\r<BR>\n" );
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document.write( "R(x) = Revenue = -0.5(x-130)^2 + 8450<BR>\n" );
document.write( "C(x) = Cost = 50x+250\r<BR>\n" );
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document.write( "Profit = Revenue - Cost<BR>\n" );
document.write( "Profit = (Amount taken in) - (Amount spent)<BR>\n" );
document.write( "P(x) = R(x) - C(x)<BR>\n" );
document.write( "P(x) = [ R(x) ] - [ C(x) ]<BR>\n" );
document.write( "P(x) = [ -0.5(x-130)^2 + 8450 ] - [ 50x+250 ]<BR>\n" );
document.write( "P(x) = -0.5(x^2-260x+16900) + 8450 - (50x+250)<BR>\n" );
document.write( "P(x) = -0.5x^2+130x-8450 + 8450 - 50x-250<BR>\n" );
document.write( "P(x) = -0.5x^2+(130x-50x)+(-8450 + 8450-250)<BR>\n" );
document.write( "P(x) = -0.5x^2 + 80x - 250\r<BR>\n" );
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document.write( "This plots a parabola which opens downward; hence, it produces a highest point (aka vertex) to quickly visually convey the max profit. <BR>\n" );
document.write( "At the end of the day, this is what the boss cares about (assuming they don't want to bother with the intricate details of the math).\r<BR>\n" );
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document.write( "Side note: The breakeven points occur at the x intercepts when P(x) = 0. This is when the revenue and cost are equal.<BR>\n" );
document.write( "Another note: The term \"sales\" is often used interchangeably with \"revenue\". Example: \"The company's sales last year were $100 million\". \r<BR>\n" );
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document.write( "Question 2) What is the domain of P(x)?\r<BR>\n" );
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document.write( "x = number of items sold\r<BR>\n" );
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document.write( "It doesn't make sense to sell a negative amount of items, nor to sell a fractional amount of items.<BR>\n" );
document.write( "The smallest x can be is x = 0<BR>\n" );
document.write( "The largest is x = 170 because this is the manufacturing plant's max capacity\r<BR>\n" );
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document.write( "Therefore, the domain is the set of integers x such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+x+%3C=+170 ALIGN=MIDDLE  ALT=\"0+%3C=+x+%3C=+170\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x is an integer between 0 and 170, inclusive of both endpoints.\r<BR>\n" );
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document.write( "We can express this in interval notation to say [0, 170].<BR>\n" );
document.write( "The square brackets mean to include each endpoint.\r<BR>\n" );
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document.write( "You could also use roster notation to write {0, 1, 2, 3, ..., 168, 169, 170}<BR>\n" );
document.write( "The triple dots tell the reader to keep the pattern going of adding 1 each time until reaching 170. This is of course much preferable instead of writing literally every integer from 0 to 170.\r<BR>\n" );
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document.write( "Question 3) The company can choose to produce either 80 or 90 items. What is their profit for each case, and which level of production should they choose?\r<BR>\n" );
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document.write( "Plug x = 80 into the P(x) function and compute.<BR>\n" );
document.write( "P(x) = -0.5x^2 + 80x - 250<BR>\n" );
document.write( "P(80) = -0.5(80)^2 + 80(80) - 250<BR>\n" );
document.write( "P(80) = 2950<BR>\n" );
document.write( "Selling 80 items yields a profit of $2950\r<BR>\n" );
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document.write( "Repeat for x = 90<BR>\n" );
document.write( "P(x) = -0.5x^2 + 80x - 250<BR>\n" );
document.write( "P(90) = -0.5(90)^2 + 80(90) - 250<BR>\n" );
document.write( "P(90) = 2900<BR>\n" );
document.write( "Selling 90 items yields a profit of $2900, which is less than the $2950 found earlier.\r<BR>\n" );
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document.write( "The company should produce and sell 80 items.\r<BR>\n" );
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document.write( "Question 4) Can you explain, from our model, why the company makes less profit when producing 10 more units?\r<BR>\n" );
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document.write( "The equation y = -0.5x^2 + 80x - 250 is in the format y = ax^2+bx+c<BR>\n" );
document.write( "The vertex is (h,k)<BR>\n" );
document.write( "h = -b/(2a)<BR>\n" );
document.write( "h = -80/(2(-0.5))<BR>\n" );
document.write( "h = 80\r<BR>\n" );
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document.write( "The vertex occurs at h = 80, meaning the highest value P(x) happens when x = 80 items are sold. <BR>\n" );
document.write( "Anything beyond this point will have the profit trend downhill.<BR>\n" );
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