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document.write( "(1) Two circles of radius 1 and 4 units are tangent\r\n" );
document.write( "to each other externally. the length of the common \r\n" );
document.write( "tangent line between the points of tangency is \r\n" );
document.write( "closest to which of the following?\r\n" );
document.write( "(a) 4.8 (b) 4.5 (c) 3.9 (d) 3.6 \r\n" );
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document.write( "Draw Circle A with radius 1 externally tangent to \r\n" );
document.write( "Circle B with radius 4.\r\n" );
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document.write( "Let CD be their common tangent, where CD is tangent \r\n" );
document.write( "to circle A at C and tangent to circle B at D.\r\n" );
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document.write( "Connect the two centers with segment AB\r\n" );
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document.write( "Draw radii AC and BD.\r\n" );
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document.write( "AC and BD are parallel because they are both\r\n" );
document.write( "perpendicular to CD.\r\n" );
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document.write( "Draw AE parallel and equal to to CD where E is on BD.\r\n" );
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document.write( "Here is the figure, I can't put in the letters here but \r\n" );
document.write( "you can on your paper:\r\n" );
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document.write( "ABE is a right triangle.\r\n" );
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document.write( "AB is its hypotenuse and equals 1+4 or 5\r\n" );
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document.write( "AC=DE=1, and BD = 4, so BE = 3\r\n" );
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document.write( "By the Pythagorean theorem\r\n" );
document.write( "BE² + AE² = AB²\r\n" );
document.write( "3² + AE² = 5²\r\n" );
document.write( "9 + AE² = 25\r\n" );
document.write( " AE² = 16\r\n" );
document.write( " AE = 4\r\n" );
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document.write( "and CD = 4 since CD=AE and AE=4\r\n" );
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document.write( "Therefore the common tangent line is 4.\r\n" );
document.write( "So the answer is (c) because 3.9 is \r\n" );
document.write( "nearest of the choices to 4.\r\n" );
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document.write( "------------------------------------- \r\n" );
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document.write( "(2)If the diameter of a cylindrical jar is \r\n" );
document.write( "increased by 25% without altering the volume \r\n" );
document.write( "then its height must decrease by:\r\n" );
document.write( "(a) 10% (b)25% (c)36% (d)50% \r\n" );
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document.write( "Let the answer be p% \r\n" );
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document.write( "Volume of a cylinder = p(radius)²(height)\r\n" );
document.write( "Radius = Diameter/2\r\n" );
document.write( "Volume of cylinder = p(diameter/2)²(height) = p(diameter)²height/4\r\n" );
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document.write( "Diameter before = d\r\n" );
document.write( "Diameter after = d + .25d = 1.25d\r\n" );
document.write( "Height before = h\r\n" );
document.write( "height after = h - .01ph = h(1-.01p)\r\n" );
document.write( "Volume before = p(d)²h/4\r\n" );
document.write( "Volume after = p(1.25d)²[h(1-.01p)]/4\r\n" );
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document.write( "Volume after = Volume before\r\n" );
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document.write( "p(1.25d)²[h(1-.01p)]/4 = p(d)²h/4\r\n" );
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document.write( "p(1.25)²d²h(1-.01p)/4 = pd²h/4\r\n" );
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document.write( "p(1.5625)d²h(1-.01p)/4 = pd²h/4\r\n" );
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document.write( "Multiply both sides by 4\r\n" );
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document.write( "p(1.5625)d²h(1-.01p) = pd²h\r\n" );
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document.write( "Divide both sides by pd²h\r\n" );
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document.write( "1.5625(1-.01p) = 1\r\n" );
document.write( "1.5625-.015625p = 1\r\n" );
document.write( " -.015625p = -.5625\r\n" );
document.write( " p = (-.5625)/(-.015625)\r\n" );
document.write( " p = 36 \r\n" );
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document.write( "So the height decreased by 36%, choice (c) \r\n" );
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document.write( "----------------------------------------\r\n" );
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document.write( "Interesting note: for some reason, statistics have shown that \r\n" );
document.write( "multiple-choice test-makers make the correct choice\r\n" );
document.write( "(c) more often than (a), (b), or (d). So if a student does\r\n" );
document.write( "not know an answer and cannot eliminate (c), and must guess, \r\n" );
document.write( "they should always guess (c), because their chances of guessing \r\n" );
document.write( "correctly is greater. You will notice that both correct choices\r\n" );
document.write( "here were indeed (c), bearing out the theory. :-)\r\n" );
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document.write( "Edwin
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
