document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174834>Question 1174834</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−110)2+6,050. The maximum capacity of the company is 150 items.<BR>\n" );
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!\r<BR>\n" );
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document.write( "Answers to some of the questions are given below so that you can check your work.\r<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)=\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 60 or 70 items. What is their profit for each case, and which level of production should they choose?<BR>\n" );
document.write( "Profit when producing 60 items =  <BR>\n" );
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document.write( "Profit when producing 70 items = <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? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #800902 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=800902\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the cost function is c(x) = 50x + 250.\r<BR>\n" );
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document.write( "the revenue function is r(x) = -.5 * (x - 110) ^ 2 + 6050.\r<BR>\n" );
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document.write( "simplify this to get r(x) = -.5 * (x^2 - 220x + 12100) + 6050.\r<BR>\n" );
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document.write( "simplify further to get r(x) = -.5x^2 + 110x - 6050 + 6050.\r<BR>\n" );
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document.write( "this results in r(x) = -.5x^2 + 110x.\r<BR>\n" );
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document.write( "the profit function is p(x) = r(x) - c(x).\r<BR>\n" );
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document.write( "you get p(x) = r(x) - c(x) = -.5x^2 + 110x - (50x + 250).\r<BR>\n" );
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document.write( "the result is p(x) = -.5x^2 + 60x - 250.\r<BR>\n" );
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document.write( "your questions with answers are shown below.\r<BR>\n" );
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document.write( "QUSTION NUMBER 1:\r<BR>\n" );
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document.write( "Assuming that the company sells all that it produces, what is the profit function?\r<BR>\n" );
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document.write( "the profit function is p(x) = -.5x^2 + 60x - 250.\r<BR>\n" );
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document.write( "it is the result of the revenue function minus the cost function.\r<BR>\n" );
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document.write( "QUESTION NUMBER 2:\r<BR>\n" );
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document.write( "What is the domain of P(x)?\r<BR>\n" );
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document.write( "the value of x represents the number of items built and sold.\r<BR>\n" );
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document.write( "since the number of items built and sold can't be less than 0, than the minimum value of x is 0.\r<BR>\n" );
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document.write( "therefore, the domain of r(x) and c(x) and p(x) requires all values of x to be greater than or equal to 0.\r<BR>\n" );
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document.write( "when x = 0, r(x) = -.5x^2 + 110x becomes r(x) = 0.<BR>\n" );
document.write( "this is ok, since you never want the revenue to become less than 0, so the value of x can be equal to 0 without violating that requirement.\r<BR>\n" );
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document.write( "when x = 0, c(x) = 50x + 250 becomes c(x) = 250.<BR>\n" );
document.write( "250 becomes the fixed cost.<BR>\n" );
document.write( "it costs the company that much regardless of how many items they build and sell.<BR>\n" );
document.write( "the incremental cost per item built is equal to 50.<BR>\n" );
document.write( "the incremental cost changes depending on the number of items built.\r<BR>\n" );
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document.write( "when x = 0, p(x) = -.5x^2 + 60x - 250 = -250.<BR>\n" );
document.write( "the profit is negative 250.<BR>\n" );
document.write( "a negative profit is a loss.<BR>\n" );
document.write( "when 0 items are built and sold, the company takes a loss of 250.<BR>\n" );
document.write( "that's to be expected since there is a fixed cost of 250 that remains the same regardless of how many items are built and sold.\r<BR>\n" );
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document.write( "the breakeven point is when p(x) = 0.<BR>\n" );
document.write( "when p(x) = 0, the revenue is equal to the cost.<BR>\n" );
document.write( "to solve for that, set p(x) = 0 in the equation of p(x) = -.5x^2 + 60x - 250.<BR>\n" );
document.write( "the equation becomes -.5x^2 + 60x - 250 = 0<BR>\n" );
document.write( "factor that quadratic equation to get:<BR>\n" );
document.write( "x = 4.3223563716998 or x = 115.6776436283\r<BR>\n" );
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document.write( "QUESTION NUMBER 3:\r<BR>\n" );
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document.write( "Does calculating P(x) make sense when x=−10 or x=1,000?\r<BR>\n" );
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document.write( "x represents the number of units built and sold.<BR>\n" );
document.write( "as such, the value of x cannot be negative.<BR>\n" );
document.write( "therefore x = -10 doesn't make any sense.\r<BR>\n" );
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document.write( "when x = 1000, the revenue is negative.<BR>\n" );
document.write( "as such, the value of x cannot be negative.<BR>\n" );
document.write( "therefore x = 1000 doesn't make any sense.\r<BR>\n" );
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document.write( "QUESTION NUMBER 4:\r<BR>\n" );
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document.write( "The company can choose to produce either 60 or 70 items. <BR>\n" );
document.write( "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 60 items = 1550.<BR>\n" );
document.write( "Profit when producing 70 items = 1500.\r<BR>\n" );
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document.write( "QUESTION NUMBER 5:\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?\r<BR>\n" );
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document.write( "the profit is less when 70 units are built and sold because the costs per unit are rising faster than the revenue per unit.\r<BR>\n" );
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document.write( "this can be seen in the graph.\r<BR>\n" );
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document.write( "the revenue increase from 60 units to 70 units is equal to 450.<BR>\n" );
document.write( "the cost increase from 60 units to 70 units is equal to 500.\r<BR>\n" );
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document.write( "here's the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/021702.jpg\" >\r<BR>\n" );
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document.write( "here's the excel spreadsheet that shows the key numbers.\r<BR>\n" );
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document.write( "i'll be available to answer any questions you might have about this.\r<BR>\n" );
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document.write( "on the graph, the green is the revenue function, the blue is the profit function, and the red is the cost function.\r<BR>\n" );
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document.write( "theo\r<BR>\n" );
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document.write( " </SPAN>\n" );
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