document.write( "Question 1204820: A nuclear accident is reported in a power plant in an isolated reservation. The location of the power plant and surrounding area is indicated using the following x-y coordinate system, where the power plant is spotted at the origin of the coordinate system.\r
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\n" ); document.write( "\n" ); document.write( "A straight road runs across the reservation as indicated in the dotted line, that intersects the x and y coordinate axis at 7 km due east (A) and 1.4 km due north (B) from the origin. Scientists are required to set up a camp, adjacent to the road between points A and B, where the radiation level is minimum. Further, the radiation level at a particular location on the first quadrant of the coordinate system was identified as the following function of distances,\r
\n" ); document.write( "\n" ); document.write( "Radiation level = 6x+5y²\r
\n" ); document.write( "\n" ); document.write( "where x and y are the distance along the x-axis and the y-axis respectively.\r
\n" ); document.write( "\n" ); document.write( "a) Formulate the optimization model for the problem.
\n" ); document.write( "b) Identify the appropriate optimization technique with justification.
\n" ); document.write( "a) Hence obtain a suitable location for a campsite along the road(assume a position on the road), where the radiation level is minimum. \n" ); document.write( "

Algebra.Com's Answer #841331 by ikleyn(52778) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "A nuclear accident is reported in a power plant in an isolated reservation. The location of the power plant
\n" ); document.write( "and surrounding area is indicated using the following x-y coordinate system, where the power plant is spotted
\n" ); document.write( "at the origin of the coordinate system.\r
\n" ); document.write( "\n" ); document.write( "A straight road runs across the reservation as indicated in the dotted line, that intersects the x and y
\n" ); document.write( "coordinate axis at 7 km due east (A) and 1.4 km due north (B) from the origin. Scientists are required
\n" ); document.write( "to set up a camp, adjacent to the road between points A and B, where the radiation level is minimum.
\n" ); document.write( "Further, the radiation level at a particular location on the first quadrant of the coordinate system
\n" ); document.write( "was identified as the following function of distances,
\n" ); document.write( "Radiation level = 6x+5y²
\n" ); document.write( "where x and y are the distance along the x-axis and the y-axis respectively.
\n" ); document.write( "a) Formulate the optimization model for the problem.
\n" ); document.write( "b) Identify the appropriate optimization technique with justification.
\n" ); document.write( " $\"highlight%28cross%28a%29%29\"$ c) Hence obtain a suitable location for a campsite along the road(assume a position on the road), where the radiation level is minimum.
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\n" ); document.write( "\n" ); document.write( "In this problem, the required steps and technique are as described below.\r
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\n" ); document.write( "\n" ); document.write( "(1)   Write an equation of the straight line defined by given points.\r
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\n" ); document.write( "\n" ); document.write( "(2)   From this equation, express x as a function of y.\r
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\n" ); document.write( "\n" ); document.write( "(3)   substitute this expression into the function describing the radiation level.
\n" ); document.write( "         It will give a radiation function on the road as a function of one variable y.\r
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\n" ); document.write( "\n" ); document.write( "(4)   The last step is to find the minimum of radiation along the road using standard method of Calculus
\n" ); document.write( "         (or more simple Algebra method, if we are lucky).\r
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\n" ); document.write( "\n" ); document.write( "Below is an implementation of this general plan.\r
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document.write( "(1)  The road passes throw points A= (7,0) and B= (0,1.4).  Hence, the slope is  m =  =  =  = -0.2.\r\n" );
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document.write( "     Thus an equation of the line is  y = -0.2(x-7),  or  y = .    (1)\r\n" );
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document.write( "     Step (1) is complete.\r\n" );
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document.write( "(2)  From equation (1)  5y = -x+7,  x = 7-5y.    (2)\r\n" );
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document.write( "     Step (2) is complete.\r\n" );
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document.write( "(3)  Substituting x from expression (2) into the radiation function, we get the radiation function on the road in the form\r\n" );
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document.write( "         R(y) =  = .    (3)\r\n" );
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document.write( "     Step (3) is complete.  We have now the radiation function on the road as a quadratic function of one coordinate y.\r\n" );
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document.write( "(4)  In some sense, we are really lucky. Since function R(y) is quadratic (= parabola), we can find its minimum value \r\n" );
document.write( "     using Algebra only,  without using Calculus. Extremum of R(y) (actually, the minimum) is achieved at the point \r\n" );
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document.write( "            = \"  \" =  =  = 3,\r\n" );
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document.write( "     and the level of radiation there is  R(3) =  = -3.\r\n" );
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document.write( "         +-------------------------------------------------------------------------------+\r\n" );
document.write( "         |    By the way, the fact that the radiation level is negative, tells us        |\r\n" );
document.write( "         |    that this function R(y) is not applicable there - see my discussion below. |\r\n" );
document.write( "         +-------------------------------------------------------------------------------+\r\n" );
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document.write( "     \r\n" );
document.write( "     But notice that this point with y-coordinate  = 3 is out of the working interval [0,1.4] for \"y\"\r\n" );
document.write( "     in the first quadrant.  Actually, the point with coordinate  = 3 lies in the second quadrant.\r\n" );
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document.write( "     Regarding quadrant 2, we have no expression for the radiation function there;  according to the problem, it is not defined\r\n" );
document.write( "     there - so, we can not move in the second quadrant and can not develop our reasonings there.\r\n" );
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document.write( "     It means that we should look at values of the function R(y) at the endpoints of the road in first quadrant,\r\n" );
document.write( "     which are y= 0  and y= 1.4.\r\n" );
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document.write( "          At y= 0    we have R(0)   = according to (3) =  = 42.\r\n" );
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document.write( "          At y= 1.4  we have R(1.4) = according to (3) =  = 9.8.\r\n" );
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document.write( "     From it, we conclude that along the road in the first quadrant, the radiation level has the minimum\r\n" );
document.write( "     at the point y= 1.4, which corresponds to point B.\r\n" );
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document.write( "So, the camp should be located at point B, according to the given restrictions/requirements.\r\n" );
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document.write( "ANSWER.  The camp should be located at point B.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "What is strange in this problem from the common sense of view is that the radiation level function\r
\n" ); document.write( "\n" ); document.write( "has this form 6x + 5y^2.\r
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\n" ); document.write( "\n" ); document.write( "From the common sense of view, this function should depend on the radius only,
\n" ); document.write( "(on the distance from the origin) and should monotonically decrease from the origin,
\n" ); document.write( "if an accident happened at the origin.\r
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\n" ); document.write( "\n" ); document.write( "So, any physicist (and any person having common sense) will be confused / (perplexed)
\n" ); document.write( "by such formulation. But, from the other side, it is a Math problem, so perhaps we should not
\n" ); document.write( "ask such questions, at all.\r
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\n" ); document.write( "\n" ); document.write( "Nevertheless, keep it in mind (and tell it to your professor/boss) that from the side, it looks ABSURDIST.\r
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\n" ); document.write( "\n" ); document.write( "Do not distribute (do not spread) it in the Internet so as not to embarrass yourself.\r
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\n" ); document.write( "\n" ); document.write( "Having such \"radiation level function\", the best location to place a camp is the epicenter at the origin, \r
\n" ); document.write( "\n" ); document.write( "because the radiation level is ZERO there, meaning that there is no radiation at the camp.\r
\n" ); document.write( "\n" ); document.write( "Surely, it is very ludicrous answer - but it is correct, in the context of given radiation function.\r
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\n" ); document.write( "\n" ); document.write( "Every day at this forum, I see incoming \"problems\" that contradict
\n" ); document.write( "to common sense or are incorrect, so it just does not surprise me.\r
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\n" ); document.write( "\n" ); document.write( "My duty is only to warn the customers about deficiencies that I see in their problems.\r
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