A semielliptic archway has a height of 20ft and a width of 50ft. Can a truck 14 ft high and 10 ft wide drive under the archway without going into the other lane?
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document.write( "Here is the drawing of the archway with the center of the\r\n" );
document.write( "highway at (0,0).\r\n" );
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document.write( "Since the center of the highway is at the origin. Let's \r\n" );
document.write( "imagine the left side of the truck being right on the center \r\n" );
document.write( "line of the highway, just barely in the right lane. \r\n" );
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document.write( "Now let's draw a rectangle 10 feet wide and 14 feet high to \r\n" );
document.write( "represent the back of the truck. Its corners will be (0,0), \r\n" );
document.write( "(0,14), (10,0), and (10,14): \r\n" );
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document.write( "So from the looks of the graph, the truck will fit nicely \r\n" );
document.write( "under there, and that we can also move the truck over to \r\n" );
document.write( "the right a foot or two, so that it won't be sitting right \r\n" );
document.write( "on the line in the middle of the road.\r\n" );
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document.write( "However, your teacher doesn't want you to just draw it and \r\n" );
document.write( "look and see without doing any calculations. We have to make \r\n" );
document.write( "sure that the upper right hand corner of the truck, (10,14), \r\n" );
document.write( "is below the semi-elliptical archway.\r\n" );
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document.write( "So let's get the equation of the whole ellipse, which is this:\r\n" );
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document.write( "Its major axis is 50, so it has semi-major axis 
. \r\n" );
document.write( "Its semi-minor axis is 20, so 
So the equation of \r\n" );
document.write( "the whole ellipse, using the equation \r\n" );
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document.write( "is\r\n" );
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document.write( "Now to find out how high the archway is above the highway at \r\n" );
document.write( "the right edge of the truck. To do this we substitute 
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document.write( "into the ellipse equation and solve for y to find the height of \r\n" );
document.write( "the archway above the highway at the right edge of the truck\r\n" );
document.write( "when its left edge is right on the center line:\r\n" );
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document.write( "Multiply thru by 400:\r\n" );
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document.write( "We take the positive square root since\r\n" );
document.write( "we are only concerned with the top\r\n" );
document.write( "half of the ellipse:\r\n" );
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document.write( "so y = about 18.3 feet.\r\n" );
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document.write( "So since the truck is only 14 feet tall,\r\n" );
document.write( "it will fit there with 4.3 feet between the \r\n" );
document.write( "top of the right edge of the back of the truck\r\n" );
document.write( "and the top of the archway.\r\n" );
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document.write( "So the answer is \"yes\". And we can move the\r\n" );
document.write( "truck right a bit so it won't be right on the \r\n" );
document.write( "center line.\r\n" );
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document.write( "Edwin
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