document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=380855>Question 380855</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Worker efficiency. In a study of worker efficiency at Wong<BR>\n" );
document.write( "Laboratories it was found that the number of components<BR>\n" );
document.write( "assembled per hour by the average worker t hours after<BR>\n" );
document.write( "starting work could be modeled by the formula<BR>\n" );
document.write( "N(t) = 3t^3 + 23t^2 + 8t<BR>\n" );
document.write( "a) Rewrite the formula by factoring the right-hand side<BR>\n" );
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document.write( "b) Use the factored version of the formula to find N(3).<BR>\n" );
document.write( "c) What is the time at which the workers are most efficient.<BR>\n" );
document.write( "d) What is the maximum number of components assembled per<BR>\n" );
document.write( "hour during an 8-hour shift. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #270341 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=270341\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula you gave is N(t) = 3t^3 + 23t^2 + 8t\r<BR>\n" );
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document.write( "I'll make x = t so this can be graphed and referenced easily.\r<BR>\n" );
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document.write( "the equation becomes n(x) = 3x^3 + 23x^2 + 8x\r<BR>\n" );
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document.write( "the graph of this formula is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-20%2C30%2C-30%2C600%2C3x%5E3%2B23x%5E2%2B8x%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-20%2C30%2C-30%2C600%2C3x%5E3%2B23x%5E2%2B8x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the formula as given doesn't make any sense.\r<BR>\n" );
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document.write( "there is no maximum when x is greater than 0.\r<BR>\n" );
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document.write( "x cannot be less than 0 so any values on the graph where x is less than 0 are to be ignored.\r<BR>\n" );
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document.write( "your graph would make more sense if the equation was:\r<BR>\n" );
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document.write( "N(t) = -3t^3 + 23t^2 + 8t which I would translate to:\r<BR>\n" );
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document.write( "n(x) = -3x^3 + 23x^2 + 8x.\r<BR>\n" );
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document.write( "that graph is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-20%2C30%2C-30%2C600%2C-3x%5E3%2B23x%5E2%2B8x%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-20%2C30%2C-30%2C600%2C-3x%5E3%2B23x%5E2%2B8x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now you have a maximum when x is greater than 0 which is more realistic.\r<BR>\n" );
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document.write( "I'll solve for n(x) = -3x^3 + 23x^2 + 8x.\r<BR>\n" );
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document.write( "x can be factored out to get:\r<BR>\n" );
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document.write( "n(x) = x * (-3x^2 + 23x + 8)\r<BR>\n" );
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document.write( "(-3x^2 + 23x + 8) factors out to be (3x+1) * (-x+8)\r<BR>\n" );
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document.write( "the completely factored equation becomes:\r<BR>\n" );
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document.write( "n(x) = x * (3x+1) * (-x+8)\r<BR>\n" );
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document.write( "when x = 0, n(x) = 0 as it should because no time worked = no components produced.\r<BR>\n" );
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document.write( "when x = 3, n(x) = (3) * (10) * (5) = 150 using the factored equation.\r<BR>\n" );
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document.write( "when x = 3, n(x) = -3(3^3) + 23*(3^2) + 8*3 = -81 + 207 + 24 = 150\r<BR>\n" );
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document.write( "you get the same answer with the factored equation and the non factored equation which confirms that the factorization is good.\r<BR>\n" );
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document.write( "I believe we need to use calculus to find the maximum / minimum point of this cubic equation.\r<BR>\n" );
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document.write( "I couldn't find any formula not using calculus on the web, so we'll use calculus.\r<BR>\n" );
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document.write( "The maximum point occurs when the derivative of the equation equals 0.\r<BR>\n" );
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document.write( "the equation is -3x^3 + 23x^2 + 8x\r<BR>\n" );
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document.write( "The derivative of this equation is -9x^2 + 46x + 8\r<BR>\n" );
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document.write( "This derivative is 0 when x = 5.279477927\r<BR>\n" );
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document.write( "That could not be factored easily, so I used the quadratic formula of:\r<BR>\n" );
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document.write( "x = (-b +/- sqrt(b^2-4ac))/(2a)\r<BR>\n" );
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document.write( "When x = 5.279477927, n(x) = 241.8493507\r<BR>\n" );
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document.write( "I'll add a line at n(x) = 241.8493507 to the graph of the equation and it should confirm that is the maximum point.\r<BR>\n" );
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document.write( "the graph looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-20%2C30%2C-30%2C600%2C-3x%5E3%2B23x%5E2%2B8x%2C241.8493507%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-20%2C30%2C-30%2C600%2C-3x%5E3%2B23x%5E2%2B8x%2C241.8493507%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The graph confirms the maximum point.\r<BR>\n" );
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document.write( "Since x represents t which represents the number of hours after starting work, then the maximum amount of output per hour occurs 5.279477927 hours after work starts.\r<BR>\n" );
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document.write( "The maximum amount of output per hour during the 8 hour shift becomes 241.8493507 units per hour which occurs at that time.\r<BR>\n" );
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document.write( "Best I can do in the short period of time allotted to the solving of this problem.\r<BR>\n" );
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document.write( "I'm not a calculus guru, but I know a little bit about it, and the little bit I know was helpful in this case.\r<BR>\n" );
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document.write( "If you were not to use calculus, then I don't really know how to find the maximum point of the cubic equation other than by iteration.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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