document.write( "Question 255987: A special stunt is being designed for an air show. A replica of the St. Louis arch has been built for stunt planes to fly under. The height of the arch is 160ft. and it is 100ft wide at the base. The pilot wants to fly his plane through the arch with the wings parallel to the ground for the opening stunt. The wingspan of the plane is 39ft and he needs a 2ft clearance on each side of the plane from the arch for safety purposes.\r
\n" ); document.write( "\n" ); document.write( "I'm supposed to matrices here, but I don't understand how to place in here other than the use of 3x1... \n" ); document.write( "
Algebra.Com's Answer #188440 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "You didn't say what you wanted to find.  But I'll assume you want\r\n" );
document.write( "to find the maximum height above the base that the plane could\r\n" );
document.write( "fly parallel to the ground and clear the archway by 2 feet on\r\n" );
document.write( "each side.\r\n" );
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document.write( "To make things easier I'm going to work in units of 10 feet\r\n" );
document.write( "instead of 1 foot.  It's just a matter of moving the decimal.\r\n" );
document.write( "So 160 ft will be represented by 16 units and 100 feet by 10\r\n" );
document.write( "units.\r\n" );
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document.write( "I'll assume the arch is a parabola with a maximum point at (0,16)\r\n" );
document.write( "with x intercepts at (-5,0) and (5,0), so that they will be 10 units\r\n" );
document.write( "(100 feet) apart.\r\n" );
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document.write( "You say you want to use matrices.  So we will use the quadratic\r\n" );
document.write( "equation\r\n" );
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document.write( "\"y=ax%5E2%2Bbx%2Bc\"\r\n" );
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document.write( "We will substitute those three points in that equation:\r\n" );
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document.write( "Substitute (x,y) = (0,16)\r\n" );
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document.write( "\"16=a%280%29%5E2%2Bb%280%29%2Bc\"\r\n" );
document.write( "\"16=0%2B0%2Bc\"\r\n" );
document.write( "\"16=c\"\r\n" );
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document.write( "Substitute (x,y) = (5,0)\r\n" );
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document.write( "\"0=a%285%29%5E2%2Bb%285%29%2Bc\"\r\n" );
document.write( "\"0=25a%2B5b%2Bc\"\r\n" );
document.write( "\"25a%2B5b%2Bc=0\"\r\n" );
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document.write( "Substitute (x,y) = (-5,0)\r\n" );
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document.write( "\"0=a%28-5%29%5E2%2Bb%28-5%29%2Bc\"\r\n" );
document.write( "\"0=25a-5b%2Bc\"\r\n" );
document.write( "\"25a-5b%2Bc=0\"\r\n" );
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document.write( "So we have the system of equations\r\n" );
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document.write( "\"system%28c=16%2C25a%2B5b%2Bc=0%2C25a-5b%2Bc=0%29\"\r\n" );
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document.write( "You can solve that by matrices or by any method you\r\n" );
document.write( "want and get \r\n" );
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document.write( "\"a=-16%2F25+=-.64\", \"b=0\", \"c=16\"\r\n" );
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document.write( "I'll assume you know how. If you don't post again\r\n" );
document.write( "asking how.\r\n" );
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document.write( "So the equation of the arch is\r\n" );
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document.write( "\"y=-0.64x%2B16\"\r\n" );
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document.write( "Its graph is this:\r\n" );
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document.write( "\"graph%28266.67%2C400%2C-6%2C6%2C-1%2C+17%2C+-0.64x%5E2%2B16%29\"\r\n" );
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document.write( "The plane is 39 feet wide and we want it to have 2 feet\r\n" );
document.write( "clearance on both sides.  So we want to know at what height\r\n" );
document.write( "the graph is 39+2+2 or 43 feet wide.  Half of that 43 feet will\r\n" );
document.write( "be on the right of the y-axis and the other half on the left, so\r\n" );
document.write( "we want to find the two points on that parabola which have \r\n" );
document.write( "x-coordinate of ±half of 43 feet, or ±21.5 feet. That will\r\n" );
document.write( "be ±2.15 of our units. so we substitute x=±2.15 into the\r\n" );
document.write( "equation.  I'll just use the positive, because it'll be\r\n" );
document.write( "the same since we square the x. \r\n" );
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document.write( "\"y=-0.64x%5E2%2B16\"\r\n" );
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document.write( "\"y=-0.64%282.15%29%5E2%2B16\"\r\n" );
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document.write( "\"y=13.0416\"\r\n" );
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document.write( "That's 13.0416 ten-foot units or 130.416 feet\r\n" );
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document.write( "In other words, to clear the edges of the arch by 2 feet,\r\n" );
document.write( "the plane will have to fly 130.416 feet above the base.\r\n" );
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document.write( "I'll draw a horizontal line across the arch to show where\r\n" );
document.write( "that will be:\r\n" );
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document.write( "Comment:\r\n" );
document.write( "Whoever made up this problem should have made the height of\r\n" );
document.write( "the arch the same as the width of the base if he wanted the\r\n" );
document.write( "arch to look like the one in St. Louis, for that one is 630 feet\r\n" );
document.write( "high and 630 feet across the base and it would look more like \r\n" );
document.write( "this:\r\n" );
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document.write( "Edwin
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