document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1033516>Question 1033516</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Q5 A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/ sec.<BR>\n" );
document.write( "a. How fast is the top of the ladder sliding down the wall then?<BR>\n" );
document.write( "b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?<BR>\n" );
document.write( "c. At what rate is the angle between the ladder and the ground changing then?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #648144 by </B><A HREF=/tutors/aboutme.mpl?userid=addingup><B>addingup(3677)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=addingup><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=addingup><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=648144\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let x be the distance between the base of the wall and the base of the ladder.  Let y be the height the top of the ladder is off the ground.  The given is <BR>\n" );
document.write( "that dx/dt = +5 at the moment x = 12.  The length of the ladder, 13 feet, is a constant, since its length doesn't change.<BR>\n" );
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document.write( "(a)  How fast is the top of the ladder sliding down the wall then? <BR>\n" );
document.write( "x^2+y^2 = 13^2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%28dx%2Fdt%29%2B2y%28dy%2Fdt%29+=+0 ALIGN=MIDDLE  ALT=\"2x%28dx%2Fdt%29%2B2y%28dy%2Fdt%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solve for dy/dt in terms of x,y, and dx/dt:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28dy%2Fdt%29+=+%28-x%2Fy%29%28dy%2Fdt%29%0D%0AThe+problem+tells+us+that+right+now+x+=+12%2C+dx%2Fdt+=+5.+Find+y+by+setting+x+to+12%3A%0D%0A%7B%7B%7Bx%5E2%2By%5E2+=+13%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > use your calculator and you get y = 5. Therefore:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Let+x+be+the+distance+between+the+base+of+the+wall+and+the+base+of+the+ladder.++Let+y+be+the+height+the+top+of+the+ladder+is+off+the+ground.++The+given+is+%0D%0Athat+dx%2Fdt+=+%2B5+at+the+moment+x+=+12.++The+length+of+the+ladder%2C+13+feet%2C+is+a+constant%2C+since+its+length+doesn%27t+change.%0D%0A%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%0D%0A%28a%29++How+fast+is+the+top+of+the+ladder+sliding+down+the+wall+then%3F+%0D%0Ax%5E2%2By%5E2+=+13%5E2%0D%0A%3CIMG+SRC=%2Fcgi-bin%2Fplot-formula.mpl%3Fexpression=2x%2528dx%252Fdt%2529%252B2y%2528dy%252Fdt%2529%2B=%2B0+ALIGN=MIDDLE++ALT=%222x%2528dx%252Fdt%2529%252B2y%2528dy%252Fdt%2529%2B=%2B0%22++ DISABLED_style= %22max-width%3A90%25%3B+height%3Aauto%3B%22+%3E%0D%0ASolve+for+dy%2Fdt+in+terms+of+x%2Cy%2C+and+dx%2Fdt%3A%0D%0A%3CIMG+SRC=%2Fcgi-bin%2Fplot-formula.mpl%3Fexpression=%2528dy%252Fdt%2529%2B=%2B%2528-x%252Fy%2529%2528dy%252Fdt%2529%250D%250AThe%2Bproblem%2Btells%2Bus%2Bthat%2Bright%2Bnow%2Bx%2B=%2B12%252C%2Bdx%252Fdt%2B=%2B5.%2BFind%2By%2Bby%2Bsetting%2Bx%2Bto%2B12%253A%250D%250A%257B%257B%257Bx%255E2%252By%255E2%2B=%2B13%255E2+ALIGN=MIDDLE++ DISABLED_style= %22max-width%3A90%25%3B+height%3Aauto%3B%22+%3E+use+your+calculator+and+you+get+y+=+5.+Therefore%3A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >dy/dx = -x*dx/y*dt = (-12/5)*5 = -12 Note that the minus sign indicates the ladder is sliding down. So, the top of the ladder is sliding down at 12 ft/second<BR>\n" );
document.write( "``````````````````````````````````````````````````````````````````````<BR>\n" );
document.write( "b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing. <BR>\n" );
document.write( "Here the unknown is the rate of change of the area, dA/dt<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Let+x+be+the+distance+between+the+base+of+the+wall+and+the+base+of+the+ladder.++Let+y+be+the+height+the+top+of+the+ladder+is+off+the+ground.++The+given+is+%0D%0Athat+dx%2Fdt+=+%2B5+at+the+moment+x+=+12.++The+length+of+the+ladder%2C+13+feet%2C+is+a+constant%2C+since+its+length+doesn%27t+change.%0D%0A%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%0D%0A%28a%29++How+fast+is+the+top+of+the+ladder+sliding+down+the+wall+then%3F+%0D%0Ax%5E2%2By%5E2+=+13%5E2%0D%0A%3CIMG+SRC=%2Fcgi-bin%2Fplot-formula.mpl%3Fexpression=2x%2528dx%252Fdt%2529%252B2y%2528dy%252Fdt%2529%2B=%2B0+ALIGN=MIDDLE++ALT=%222x%2528dx%252Fdt%2529%252B2y%2528dy%252Fdt%2529%2B=%2B0%22++ DISABLED_style= %22max-width%3A90%25%3B+height%3Aauto%3B%22+%3E%0D%0ASolve+for+dy%2Fdt+in+terms+of+x%2Cy%2C+and+dx%2Fdt%3A%0D%0A%3CIMG+SRC=%2Fcgi-bin%2Fplot-formula.mpl%3Fexpression=%2528dy%252Fdt%2529%2B=%2B%2528-x%252Fy%2529%2528dy%252Fdt%2529%250D%250AThe%2Bproblem%2Btells%2Bus%2Bthat%2Bright%2Bnow%2Bx%2B=%2B12%252C%2Bdx%252Fdt%2B=%2B5.%2BFind%2By%2Bby%2Bsetting%2Bx%2Bto%2B12%253A%250D%250A%257B%257B%257Bx%255E2%252By%255E2%2B=%2B13%255E2+ALIGN=MIDDLE++ DISABLED_style= %22max-width%3A90%25%3B+height%3Aauto%3B%22+%3E+use+your+calculator+and+you+get+y+=+5.+Therefore%3A%0D%0A%3CIMG+SRC=%2Fcgi-bin%2Fplot-formula.mpl%3Fexpression=Let%2Bx%2Bbe%2Bthe%2Bdistance%2Bbetween%2Bthe%2Bbase%2Bof%2Bthe%2Bwall%2Band%2Bthe%2Bbase%2Bof%2Bthe%2Bladder.%2B%2BLet%2By%2Bbe%2Bthe%2Bheight%2Bthe%2Btop%2Bof%2Bthe%2Bladder%2Bis%2Boff%2Bthe%2Bground.%2B%2BThe%2Bgiven%2Bis%2B%250D%250Athat%2Bdx%252Fdt%2B=%2B%252B5%2Bat%2Bthe%2Bmoment%2Bx%2B=%2B12.%2B%2BThe%2Blength%2Bof%2Bthe%2Bladder%252C%2B13%2Bfeet%252C%2Bis%2Ba%2Bconstant%252C%2Bsince%2Bits%2Blength%2Bdoesn%2527t%2Bchange.%250D%250A%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%2560%250D%250A%2528a%2529%2B%2BHow%2Bfast%2Bis%2Bthe%2Btop%2Bof%2Bthe%2Bladder%2Bsliding%2Bdown%2Bthe%2Bwall%2Bthen%253F%2B%250D%250Ax%255E2%252By%255E2%2B=%2B13%255E2%250D%250A%253CIMG%2BSRC=%252Fcgi-bin%252Fplot-formula.mpl%253Fexpression=2x%252528dx%25252Fdt%252529%25252B2y%252528dy%25252Fdt%252529%252B=%252B0%2BALIGN=MIDDLE%2B%2BALT=%25222x%252528dx%25252Fdt%252529%25252B2y%252528dy%25252Fdt%252529%252B=%252B0%2522%2B%2Bstyle=%2522max-width%253A90%2525%253B%2Bheight%253Aauto%253B%2522%2B%253E%250D%250ASolve%2Bfor%2Bdy%252Fdt%2Bin%2Bterms%2Bof%2Bx%252Cy%252C%2Band%2Bdx%252Fdt%253A%250D%250A%253CIMG%2BSRC=%252Fcgi-bin%252Fplot-formula.mpl%253Fexpression=%252528dy%25252Fdt%252529%252B=%252B%252528-x%25252Fy%252529%252528dy%25252Fdt%252529%25250D%25250AThe%252Bproblem%252Btells%252Bus%252Bthat%252Bright%252Bnow%252Bx%252B=%252B12%25252C%252Bdx%25252Fdt%252B=%252B5.%252BFind%252By%252Bby%252Bsetting%252Bx%252Bto%252B12%25253A%25250D%25250A%25257B%25257B%25257Bx%25255E2%25252By%25255E2%252B=%252B13%25255E2%2BALIGN=MIDDLE%2B%2Bstyle=%2522max-width%253A90%2525%253B%2Bheight%253Aauto%253B%2522%2B%253E%2Buse%2Byour%2Bcalculator%2Band%2Byou%2Bget%2By%2B=%2B5.%2BTherefore%253A%250D%250A+ALIGN=MIDDLE++ DISABLED_style= %22max-width%3A90%25%3B+height%3Aauto%3B%22+%3Edy%2Fdx+=+-x%2Adx%2Fy%2Adt+=+%28-12%2F5%29%2A5+=+-12+Note+that+the+minus+sign+indicates+the+ladder+is+sliding+down.+So%2C+the+top+of+the+ladder+is+sliding+down+at+12+ft%2Fsecond%0D%0A%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%60%0D%0Ab.+At+what+rate+is+the+area+of+the+triangle+formed+by+the+ladder%2C+wall%2C+and+ground+changing.+%0D%0AHere+the+unknown+is+the+rate+of+change+of+the+area%2C+dA%2Fdt%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >dA/dt = 1/2 (x*dy/dt+y*dx/dt)<BR>\n" );
document.write( "Substitute with x = 12 and dx/dt = 5 and y=5 dy/dt = -12:<BR>\n" );
document.write( "dA/dt = 1//2(12*(-12)5*5)= -144+25/2 = -119/2 = 59.5 ft^2/sec<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~<BR>\n" );
document.write( "c. At what rate is the angle between the ladder and the ground changing then?<BR>\n" );
document.write( "ormula that relates the angle &#952; with the sides x and y of a right triangle: <BR>\n" );
document.write( "Answer<BR>\n" );
document.write( "1 rad/sec<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );