document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172726>Question 172726</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0395977258/algebrahomeworkh>Algebra and Trigonomety</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> 1. pipe A can fill a tank in 5 h. pipe B can fill it in 2 h less time than it takes pipe C, a drainpipe, to empty the tank. with all three pipes open, it takes 3 h to fill the tank. how long would it take pipe C to empty it?\r<BR>\n" );
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document.write( "2. elvin drove halfway from Ashton to Dover at 40 mi/h and the rest of the way at 60 mi/h. what was his average speed for the whole trip? (Hint: let the distance for the whole trip be, say, 100 mi.)\r<BR>\n" );
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document.write( "3. the reciprocal of half a number increased by half the reciprocal of the number is 1/2. find the number.\r<BR>\n" );
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document.write( "4. find two positive numbers that differ by 8 and whose reciprocals differ by 1/6. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #127567 by </B><A HREF=/tutors/aboutme.mpl?userid=ptaylor><B>ptaylor(2198)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ptaylor><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/cgi-bin/show-face.mpl?user=ptaylor><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=127567\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "1)  <BR>\n" );
document.write( "Let x=time, in hours, that it takes pipe C to empty the tank<BR>\n" );
document.write( "Then pipe C empties the tank at the rate of 1/x tank per hour<BR>\n" );
document.write( "Pipe A fills at the rate of 1/5 tank per hour<BR>\n" );
document.write( "We are told that it takes pipe B (x-2) hours to fill the tank<BR>\n" );
document.write( "So pipe B fills at the rate of 1/(x-2) tank per hour<BR>\n" );
document.write( "With all three pipes open, they fill at the rate of 1/3 tank per hour<BR>\n" );
document.write( "So our equation to solve is:<BR>\n" );
document.write( "1/5 +1/(x-2)-1/x=1/3  multiply each term by 15x(x-2)<BR>\n" );
document.write( "3x(x-2)+15x-15(x-2)=5x(x-2)  get rid of parens<BR>\n" );
document.write( "3x^2-6x+15x-15x+30=5x^2-10x subtract 5x^2 from and add 10x to both sides<BR>\n" );
document.write( "3x^2-5x^2-6x+15x-15x+30+10x=5x^2-5x^2-10x+10x collect like terms<BR>\n" );
document.write( "-2x^2+4x+30=0  divide each term by -2<BR>\n" );
document.write( "x^2-2x-15=0----------------------------quadratic in standard form and it can be factored<BR>\n" );
document.write( "(x-5)(x+3)=0<BR>\n" );
document.write( "x-5=0<BR>\n" );
document.write( "x=5 hours------------------------------answer<BR>\n" );
document.write( "and<BR>\n" );
document.write( "x+3=0<BR>\n" );
document.write( "x=-3-----------------------------disregard negative value; hours in this case are positive\r<BR>\n" );
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document.write( "CK<BR>\n" );
document.write( "1/5+1/(5-2)-1/5=1/3<BR>\n" );
document.write( "1/5+1/3-1/5=1/3<BR>\n" );
document.write( "1/3=1/3\r<BR>\n" );
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document.write( "2) <BR>\n" );
document.write( "Distance(d)=Rate(r) times Time(t) or d=rt; r=d/t and t=d/r<BR>\n" );
document.write( "Average speed=(Total distance)/(Total time)<BR>\n" );
document.write( "Let d=distance from Ashton to Dover (you can use any distance because the distance cancels out in the end.  In fact, it might be a little easier to use, say, 100 mi)<BR>\n" );
document.write( "Time required to drive the first half of the trip=(d/2)/40=d/80<BR>\n" );
document.write( "Time required to drive the second half of the trip=(d/2)/60=d/120<BR>\n" );
document.write( "Total time =d/80 + d/120=(3d+2d)/240=5d/240=d/48 hr<BR>\n" );
document.write( "So, average speed=d/(d/48)=d*(48/d)=48 mi/hr-----------------ans\r<BR>\n" );
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document.write( "3)<BR>\n" );
document.write( "Let x=the number<BR>\n" );
document.write( "The reciprocal of the number is 1/x;<BR>\n" );
document.write( "half the reciprocal of the number is 1/2x<BR>\n" );
document.write( "Half the number =x/2<BR>\n" );
document.write( "The reciprocal of half the number =2/x<BR>\n" );
document.write( "So, our equation to solve is:<BR>\n" );
document.write( "2/x (the reciprocal of half the number)+ 1/2x(half the reciprocal of the number) equals 1/2, or<BR>\n" );
document.write( "2/x +1/2x = 1/2 multiply each term by 2x<BR>\n" );
document.write( "4+1=x or<BR>\n" );
document.write( "x=5\r<BR>\n" );
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document.write( "CK<BR>\n" );
document.write( "reciprocal of half the number =2/5<BR>\n" );
document.write( "half the reciprocal of the number=2/10<BR>\n" );
document.write( "So, 2/5 + 2/10=1/2<BR>\n" );
document.write( "4/10 + 1/10=1/2<BR>\n" );
document.write( "1/2=1/2\r<BR>\n" );
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document.write( "4) <BR>\n" );
document.write( "Let x= one number<BR>\n" );
document.write( "1/x=reciprocal of the number<BR>\n" );
document.write( "Then x-8=the other number<BR>\n" );
document.write( "1/(x-8)= reciprocal<BR>\n" );
document.write( "So, our equation to solve is:<BR>\n" );
document.write( "1/(x-8) - 1/x=1/6---eq1<BR>\n" );
document.write( "(note: although x>(x-8) the reciprocal of x is smaller than the reciprocal of (x-8))\r<BR>\n" );
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document.write( "Multiply each term in eq1 above by 6x(x-8)<BR>\n" );
document.write( "6x-6(x-8)=x(x-8)  get rid of parens<BR>\n" );
document.write( "6x-6x+48=x^2-8x  subtract x^2 from and add 8x to each side<BR>\n" );
document.write( "6x-6x+48-x^2+8x=x^2-x^2-8x+8x  collect like terms<BR>\n" );
document.write( "-x^2+8x+48=0  multiply each term by -1<BR>\n" );
document.write( "x^2-8x-48=0-------------------quadratic in standard form and it can be factored:<BR>\n" );
document.write( "(x-12)(x+4)=0<BR>\n" );
document.write( "x=12-----------------------one number<BR>\n" );
document.write( "x=-4--------------------------disregard negative value<BR>\n" );
document.write( "x-8=12-8=4------------------------the other number\r<BR>\n" );
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document.write( "CK<BR>\n" );
document.write( "1/4-1/12=1/6<BR>\n" );
document.write( "3/12-1/12=1/6<BR>\n" );
document.write( "2/12=1/6<BR>\n" );
document.write( "1/6=1/6\r<BR>\n" );
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document.write( "Does this help???---------------------ptaylor\r<BR>\n" );
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document.write( " </SPAN>\n" );
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