document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=278152>Question 278152</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Not sure where to start with this one:\r<BR>\n" );
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document.write( "An orchard contains 30 apple trees, each of which yields approx 400 apples over the growing season.  The owner plans to add more trees, but the guys at Texas A&M advise that b/c of crowding, each new tree will reduce the average yield per tree by about 10 apples over the growing season.  How many trees should be added to maximize the total yield of apples, and whit is the maximum yield?\r<BR>\n" );
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document.write( "x=30<BR>\n" );
document.write( "y=400 I think, but how do I plug this into a quadratic function?  Or is that even the direction I need to go? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #202426 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=202426\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x = number of new trees to be planted.\r<BR>\n" );
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document.write( "So adding 'x' trees to the already established 30 trees gives us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2Bx ALIGN=MIDDLE  ALT=\"30%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > trees total. Now because \"each new tree will reduce the average yield per tree by about 10 apples over the growing season\", this means that each tree will produce <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400-10x ALIGN=MIDDLE  ALT=\"400-10x\"   DISABLED_style= \"max-width:90%; height:auto;\" > apples. To see why, let x=1 which means that 400-10(1)=400-10=390. So planting one new tree means that each tree only produces 390 apples, which is what we expect. Try other values of 'x' to confirm this expression.\r<BR>\n" );
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document.write( "Since we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2Bx ALIGN=MIDDLE  ALT=\"30%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" > trees which each produce <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400-10x ALIGN=MIDDLE  ALT=\"400-10x\"   DISABLED_style= \"max-width:90%; height:auto;\" > apples, this means that there will be a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2830%2Bx%29%28400-10x%29 ALIGN=MIDDLE  ALT=\"%2830%2Bx%29%28400-10x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > apples total after 'x' trees are planted.\r<BR>\n" );
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document.write( "Note: if there is a part that doesn't make sense explained above, it is wise to go over it again or ask me about it. Translation is a critical element to solving word problems.\r<BR>\n" );
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document.write( "So let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%2830%2Bx%29%28400-10x%29 ALIGN=MIDDLE  ALT=\"y=%2830%2Bx%29%28400-10x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%2830%2Bx%29%28400-10x%29 ALIGN=MIDDLE  ALT=\"y=%2830%2Bx%29%28400-10x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12000-300x%2B400x-10x%5E2 ALIGN=MIDDLE  ALT=\"y=12000-300x%2B400x-10x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-10x%5E2%2B100x%2B12000 ALIGN=MIDDLE  ALT=\"y=-10x%5E2%2B100x%2B12000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange and combine like terms.\r<BR>\n" );
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document.write( "Now the goal is to maximize 'y' which is the total number of apples. To do this, we need to find the y coordinate of the vertex. Remember that the vertex is the highest/lowest point on a graph. In this case, it is the highest point on the parabola.\r<BR>\n" );
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document.write( "In order to find the vertex, we first need to find the x-coordinate of the vertex.\r<BR>\n" );
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document.write( "To find the x-coordinate of the vertex, use this formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-b%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x=%28-b%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given formula.\r<BR>\n" );
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document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-10x%5E2%2B100x%2B12000 ALIGN=MIDDLE  ALT=\"y=-10x%5E2%2B100x%2B12000\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-10 ALIGN=MIDDLE  ALT=\"a=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=100 ALIGN=MIDDLE  ALT=\"b=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=12000 ALIGN=MIDDLE  ALT=\"c=12000\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-%28100%29%29%2F%282%28-10%29%29 ALIGN=MIDDLE  ALT=\"x=%28-%28100%29%29%2F%282%28-10%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-10 ALIGN=MIDDLE  ALT=\"a=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=100 ALIGN=MIDDLE  ALT=\"b=100\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-100%29%2F%28-20%29 ALIGN=MIDDLE  ALT=\"x=%28-100%29%2F%28-20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10 ALIGN=MIDDLE  ALT=\"-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20 ALIGN=MIDDLE  ALT=\"-20\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide.\r<BR>\n" );
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document.write( "So the x-coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Note: this means that the axis of symmetry is also <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now that we know the x-coordinate of the vertex, we can use it to find the y-coordinate of the vertex.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-10x%5E2%2B100x%2B12000 ALIGN=MIDDLE  ALT=\"y=-10x%5E2%2B100x%2B12000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-10%285%29%5E2%2B100%285%29%2B12000 ALIGN=MIDDLE  ALT=\"y=-10%285%29%5E2%2B100%285%29%2B12000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-10%2825%29%2B100%285%29%2B12000 ALIGN=MIDDLE  ALT=\"y=-10%2825%29%2B100%285%29%2B12000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-250%2B100%285%29%2B12000 ALIGN=MIDDLE  ALT=\"y=-250%2B100%285%29%2B12000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10 ALIGN=MIDDLE  ALT=\"-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-250 ALIGN=MIDDLE  ALT=\"-250\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-250%2B500%2B12000 ALIGN=MIDDLE  ALT=\"y=-250%2B500%2B12000\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500 ALIGN=MIDDLE  ALT=\"500\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12250 ALIGN=MIDDLE  ALT=\"y=12250\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So the y-coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12250 ALIGN=MIDDLE  ALT=\"y=12250\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So the vertex is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(5,12250\right)\">.\r<BR>\n" );
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document.write( "Since the y-coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12250 ALIGN=MIDDLE  ALT=\"y=12250\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that the maximum production of apples is 12,250 which occurs when 5 new trees are planted (since x=5 at the vertex). </SPAN>\n" );
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