document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=728358>Question 728358</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A man 2m tall walks horizontally at a constant rate of 1 m/s toward the base of a tower 25m tall. When the man is 10m from the tower, how fast is the angle of elevation changing if that angle is measured from the horizontal to the line joining the top of the man's head to the top of the tower? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #445395 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=445395\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this problem in a strictly accurate way requires calculus, because the rate of change of the angle is not constant.<BR>\n" );
document.write( "We could an expression for the angle as a function of time, calculate the derivative of that function, and then calculate the value of that derivative at the time the man is 10m from the tower.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Alternatively we could obtain an approximate value by calculating the average rate of changeover short period(s) of time at around that time.<BR>\n" );
document.write( " <BR>\n" );
document.write( "APPROXIMATE SOLUTION:<BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" >= angle of elevation<BR>\n" );
document.write( "when the man is 10 meters from the base of the tower,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=23%2F10 ALIGN=MIDDLE  ALT=\"tan%28theta%29=23%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" > because the horizontal from the top of the 2 meter tall man's head passes by a point 2 meters above the base of the tower and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25meters-2meters=23meters ALIGN=MIDDLE  ALT=\"25meters-2meters=23meters\"   DISABLED_style= \"max-width:90%; height:auto;\" > below the top of the tower<BR>\n" );
document.write( "(assuming the ground is horizontal between the man and the tower).<BR>\n" );
document.write( "When the man is 11 meters from the base of the tower,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=23%2F11 ALIGN=MIDDLE  ALT=\"tan%28theta%29=23%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "when the man is 9 meters from the base of the tower,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=23%2F9 ALIGN=MIDDLE  ALT=\"tan%28theta%29=23%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Approximate calculations of tangents and the corresponding angles in radians:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=23%2F9=2.555556 ALIGN=MIDDLE  ALT=\"tan%28theta%29=23%2F9=2.555556\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=1.197809 ALIGN=MIDDLE  ALT=\"theta=1.197809\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=23%2F10=2.3 ALIGN=MIDDLE  ALT=\"tan%28theta%29=23%2F10=2.3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=1.160669 ALIGN=MIDDLE  ALT=\"theta=1.160669\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The average rate of change of that angle of elevation over the second the man shortens his distance from the foot of the tower from 10m to 9m , in radians per second, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281.197809-1.160669%29%2F1=0.037140 ALIGN=MIDDLE  ALT=\"%281.197809-1.160669%29%2F1=0.037140\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In degrees per second, it would be approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.13 ALIGN=MIDDLE  ALT=\"2.13\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=23%2F11=2.090909 ALIGN=MIDDLE  ALT=\"tan%28theta%29=23%2F11=2.090909\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=1.124691 ALIGN=MIDDLE  ALT=\"theta=1.124691\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The average rate of change of that angle of elevation over the second the man shortens his distance from the foot of the tower from 11m to 10m , in radians per second, is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281.124691-1.197809%29%2F1=0.035978 ALIGN=MIDDLE  ALT=\"%281.124691-1.197809%29%2F1=0.035978\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In degrees per second, it would be approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.06 ALIGN=MIDDLE  ALT=\"2.06\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The instantaneous rate of change at thye precise time when the man is 10 meters from the base of the tower must be somewhere in between, but maybe <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%282%5Eo%29 ALIGN=MIDDLE  ALT=\"highlight%282%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > per second is accurate enough for an answer.<BR>\n" );
document.write( "Otherwise, we could average the results an come up with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280.037140%2B0.035978%29%2F2=about0.03656 ALIGN=MIDDLE  ALT=\"%280.037140%2B0.035978%29%2F2=about0.03656\"   DISABLED_style= \"max-width:90%; height:auto;\" > radians per second = about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.09%5Eo ALIGN=MIDDLE  ALT=\"2.09%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > per second<BR>\n" );
document.write( " <BR>\n" );
document.write( "WITH CALCULUS:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >= time until the man reaches the bottom of the tower, in seconds.<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >= horizontal distance between the man and the tower in meters<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=23%2Ft ALIGN=MIDDLE  ALT=\"tan%28theta%29=23%2Ft\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=tan%5E%28-1%29%28theta%29 ALIGN=MIDDLE  ALT=\"theta=tan%5E%28-1%29%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%5E-1%2823%2Ft%29 ALIGN=MIDDLE  ALT=\"tan%5E-1%2823%2Ft%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=tan%5E-1%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29=tan%5E-1%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=df%2Fdx=1%2F%281%2Bx%5E2%29 ALIGN=MIDDLE  ALT=\"df%2Fdx=1%2F%281%2Bx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the rate of change of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+as+a+function+of+%7B%7B%7Bt ALIGN=MIDDLE  ALT=\"theta+as+a+function+of+%7B%7B%7Bt\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+theta%2Fdt=%281%2F%281%2B%2823%2Ft%29%5E2%29%29%2823%29%28-1%2Ft%5E2%29=%28-23%2Ft%5E2%29%281%2F%281%2Bt%5E2%2F23%5E3%29%29=%28-23%2Ft%5E2%29%281%2F%28%28t%5E2%2B23%5E2%29%2Ft%5E2%29%29=%28-23%2Ft%5E2%29%28t%5E2%2F%28t%5E2%2B23%5E2%29%29=-23%2F%28t%5E2%2B23%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That rate of change is negative, of course, because as the time needed to reach the foot of the tower increases, the angle decreases.<BR>\n" );
document.write( "We are interested in the opposite because we are decreasing the time needed to reach the foot of the tower.<BR>\n" );
document.write( "The rate of increase of the angle as the man approaches the tower, as a function<BR>\n" );
document.write( "of time is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23%2F%28t%5E2%2B23%5E2%29=23%2F%28t%5E2%2B529%29 ALIGN=MIDDLE  ALT=\"23%2F%28t%5E2%2B23%5E2%29=23%2F%28t%5E2%2B529%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=10 ALIGN=MIDDLE  ALT=\"t=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > that rate is exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23%2F%2810%5E2%2B529%29=23%2F100%2B529=23%2F629 ALIGN=MIDDLE  ALT=\"23%2F%2810%5E2%2B529%29=23%2F100%2B529=23%2F629\"   DISABLED_style= \"max-width:90%; height:auto;\" > radians per second, with units of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E-1 ALIGN=MIDDLE  ALT=\"s%5E-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In the more familiar degrees per second, the numerical value would be exactly<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2823%2F629%29%28180%2Fpi%29%7D%7D+=+approx.+%7B%7B%7B2.09508 ALIGN=MIDDLE  ALT=\"%2823%2F629%29%28180%2Fpi%29%7D%7D+=+approx.+%7B%7B%7B2.09508\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );