document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1085831>Question 1085831</A>: <I><SPAN STYLE=\"word-break: break-all;\"> to what height above the earth's surface must a man be raised for him to see 1/4 of the earth's surface </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #699921 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=699921\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Use this as a reference<BR>\n" );
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document.write( "Let R be the radius of the earth. It doesn't matter what R is. This R value can change (say we go to another planet, the idea still holds). In the drawing, I made R = 2, but again the value for R doesn't matter.\r<BR>\n" );
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document.write( "Place point A at the origin (0,0). Place point B to be R units away from point A. So let's say B = (R, 0). Draw a circle centered at point A and that goes through point B. The equation for this circle is x^2+y^2 = R^2. Call this circle p. \r<BR>\n" );
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document.write( "Place point C at (0,R). Draw a ray from point A that extends through point C and goes on forever from there. Plot a point D such that D is on the ray but not between A and C. This point D is going to represent the person's location. Since the ray is a vertical line, we only need to worry about the y coordinate of point D. If the person's height off the ground of the planet is k, then D = (0,k+R). \r<BR>\n" );
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document.write( "We have<BR>\n" );
document.write( "A = (0,0) and D = (0,k+R)<BR>\n" );
document.write( "Find the midpoint of A and D to get E = (0,(k+R)/2)\r<BR>\n" );
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document.write( "Draw a circle centered at point E and that goes through point A, or point D. This circle will have the equation x^2 + (y - (k+R)/2)^2 = ((k+R)/2)^2. Call this circle q\r<BR>\n" );
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document.write( "The two circle equations are<BR>\n" );
document.write( "p: x^2+y^2 = R^2<BR>\n" );
document.write( "q: x^2 + (y - (k+R)/2)^2 = ((k+R)/2)^2\r<BR>\n" );
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document.write( "Subtract the equations p-q to get<BR>\n" );
document.write( "[x^2+y^2] - [x^2 + (y - (k+R)/2)^2] = R^2 - ((k+R)/2)^2<BR>\n" );
document.write( "y^2 - (y - (k+R)/2)^2 = R^2 - ((k+R)/2)^2<BR>\n" );
document.write( "y^2 - (y^2 - 2*y*(k+R)/2 + ((k+R)/2)^2) = R^2 - ((k+R)/2)^2<BR>\n" );
document.write( "y^2 - (y^2 - y*(k+R) + ((k+R)/2)^2) = R^2 - ((k+R)/2)^2<BR>\n" );
document.write( "y^2 - y^2 + y*(k+R) - ((k+R)/2)^2 = R^2 - ((k+R)/2)^2<BR>\n" );
document.write( "y*(k+R) = R^2<BR>\n" );
document.write( "y = (R^2)/(k+R)\r<BR>\n" );
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document.write( "What does this mean? Well it means that circle p and circle q cross at two points, call them F and G. Point F and point G have the same y coordinate, and that y coordinate is equal to (R^2)/(k+R)\r<BR>\n" );
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document.write( "The vertical distance from point F to point C is <BR>\n" );
document.write( "R - (R^2)/(k+R) = (R*(k+R))/(k+R) - (R^2)/(k+R)<BR>\n" );
document.write( "R - (R^2)/(k+R) = (R*k+R^2)/(k+R) - (R^2)/(k+R)<BR>\n" );
document.write( "R - (R^2)/(k+R) = (R*k+R^2-R^2)/(k+R)<BR>\n" );
document.write( "R - (R^2)/(k+R) = (R*k)/(k+R)<BR>\n" );
document.write( "R - (R^2)/(k+R) = (k*R)/(k+R)\r<BR>\n" );
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document.write( "This is the height h of the spherical cap as shown on <a  rel=nofollow HREF=\"http://mathworld.wolfram.com/SphericalCap.html\">this article</a>. Scroll to the bottom of that page and you'll see the formula <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large S_{\text{cap}} = 2*\pi*R*h\">. Also, the blue portion represents the portion that the person can see for a given height.\r<BR>\n" );
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document.write( "Let's plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h+=+%28R%2Ak%29%2F%28k%2BR%29 ALIGN=MIDDLE  ALT=\"h+=+%28R%2Ak%29%2F%28k%2BR%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which we found earlier\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large S_{\text{cap}} = 2*\pi*R*h\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large S_{\text{cap}} = 2*\pi*R*\frac{k*R}{k+R}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large S_{\text{cap}} = \frac{2*\pi*R^2*k}{k+R}\">\r<BR>\n" );
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document.write( "If we let k = x, then we can make the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%28x%29+=+%282%2Api%2AR%5E2%2Ax%29%2F%28x%2BR%29 ALIGN=MIDDLE  ALT=\"S%28x%29+=+%282%2Api%2AR%5E2%2Ax%29%2F%28x%2BR%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > allowing us to find the surface area of a spherical cap for any height x off the ground. \r<BR>\n" );
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document.write( "We're told that we want the surface area of the cap to be 1/4 of the surface area of the planet (aka sphere). So,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large S_{\text{cap}} = \frac{1}{4}*S_{\text{sphere}}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large S_{\text{cap}} = \frac{1}{4}*4*\pi*R^2\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large S_{\text{cap}} = \pi*R^2\">\r<BR>\n" );
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document.write( "We want the surface area of the cap to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2 ALIGN=MIDDLE  ALT=\"pi%2AR%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Plug this into S(x) and solve for x<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2+=+%282%2Api%2AR%5E2%2Ax%29%2F%28x%2BR%29 ALIGN=MIDDLE  ALT=\"pi%2AR%5E2+=+%282%2Api%2AR%5E2%2Ax%29%2F%28x%2BR%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2%2A%28x%2BR%29+=+2%2Api%2AR%5E2%2Ax ALIGN=MIDDLE  ALT=\"pi%2AR%5E2%2A%28x%2BR%29+=+2%2Api%2AR%5E2%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E3+=+2%2Api%2AR%5E2%2Ax-pi%2AR%5E2%2Ax ALIGN=MIDDLE  ALT=\"pi%2AR%5E3+=+2%2Api%2AR%5E2%2Ax-pi%2AR%5E2%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E3+=+pi%2AR%5E2%2Ax ALIGN=MIDDLE  ALT=\"pi%2AR%5E3+=+pi%2AR%5E2%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2AR%5E2%2Ax+=+pi%2AR%5E3 ALIGN=MIDDLE  ALT=\"pi%2AR%5E2%2Ax+=+pi%2AR%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28pi%2AR%5E3%29%2F%28pi%2AR%5E2%29 ALIGN=MIDDLE  ALT=\"x+=+%28pi%2AR%5E3%29%2F%28pi%2AR%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+R ALIGN=MIDDLE  ALT=\"x+=+R\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It turns out that if the height of the person is exactly equal to the radius of the planet, then the person will be able to see 1/4 of the sphere's surface area\r<BR>\n" );
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document.write( "The radius of the Earth is roughly R = 3959 miles<BR>\n" );
document.write( "So the person should be at a height of x = R = 3959 miles off the ground\r<BR>\n" );
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document.write( "Note: The earth isn't a perfect sphere, but I'm assuming it is just to make things a bit more simple.</font> </SPAN>\n" );
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