document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210455>Question 1210455</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Skyball is a popular sport in the Magical Realm where the Chaser of each team is tasked with catching the Silver Orb. In today’s match, Lina and Kai are the Chasers for opposing teams. Kai is floating directly above Lina when both spot the Silver Orb. In this instance, Lina’s angle of elevation to the Orb is 35°, while Kai’s angle of depression to it is 42°. Meanwhile, Orin, seated 8.2 meters directly below the Orb, watches Lina with a 28° angle of elevation. (Show your illustration)\r<BR>\n" );
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document.write( "(a) What is the distance between Lina and the Silver Orb? How about between Kai and the Orb?<BR>\n" );
document.write( "(b) Orin wants Kai’s team to win, so he magically lifts the Silver Orb 1.5 meters higher. How far is Kai from the Orb now? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852651 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5330)</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=852651\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The drawing is not to scale because of typos, but hopefully my calculations were typo-free.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C600%2C+-0.5%2C8.5%2C+-9%2C7%2C%0D%0Atriangle%280%2C6.33+%2C7.03%2C6.33%2C+7.03%2C0%29%2Clocate%287.2%2C6.5%2CA%29%2C%0D%0Atriangle%287.03%2C0+%2C7.03%2C-8.2%2C+0%2C-4.92%29%2Clocate%287.2%2C-4.7%2CB%29%2C%0D%0Arectangle%280%2C-8.2%2C+7.03%2C-4.92%29%2C%0D%0Acircle%287.03%2C0%2C0.2%29%2Clocate%287.4%2C0.2%2CS%29%0D%0Alocate%284.8%2C0.2%2COrb%29%2Clocate%283.4%2C0.2%2CSilver%29%2Carrow%285.8%2C0%2C6.8%2C0%29%2C%0D%0Acircle%287.03%2C-8.2%2C0.05%29%2Clocate%287.2%2C-8%2CO%29%2Clocate%287.6%2C-8.5%2COrin%29%2C%0D%0Acircle%280%2C-4.92%2C0.05%29%2Clocate%28-0.45%2C-3.5%2CLina%29%2Clocate%28-0.3%2C-4.7%2CL%29%2C%0D%0Aarrow%280%2C-3.8%2C0%2C-4.85%29%2Carrow%280%2C4.8%2C0%2C6.25%29%2Carrow%287.6%2C-8.7%2C7.1%2C-8.3%29%2C%0D%0Acircle%280%2C6.33%2C0.05%29%2Clocate%28-0.4%2C4.8%2CKai%29%2Clocate%28-0.3%2C6.5%2CK%29%2C%0D%0Aarc%280%2C-4.92%2C+4%2C4%2C-35%2C0%29%2Clocate%282%2C-4%2C35%5Eo%29%2C%0D%0Aarc%280%2C6.33%2C+4%2C4%2C0%2C42%29%2Clocate%282%2C6%2C42%5Eo%29%2C%0D%0Aarc%287.03%2C-8.2%2C+2%2C2%2C180%2C205%29%2Carc%287.03%2C-8.2%2C+2.3%2C2.3%2C180%2C205%29%2Clocate%285%2C-7.5%2C28%5Eo%29%2C+rectangle%280%2C-8.2%2C0.3%2C-7.9%29%2Crectangle%286.73%2C-5.22%2C7.03%2C-4.62%29%2C%0D%0Arectangle%287.03%2C6.33%2C6.73%2C6.03%29%2C%0D%0Aarrow%283.51%2C-8.75%2C0%2C-8.75%29%2Carrow%283.51%2C-8.75%2C7.03%2C-8.75%29%2C%0D%0Aarc%280%2C-4.92%2C2%2C2%2C0%2C25%29%2Carc%280%2C-4.92%2C2.2%2C2.2%2C0%2C25%29%2Clocate%283.5%2C-8.35%2CL%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Consider right triangles <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=KAS ALIGN=MIDDLE  ALT=\"KAS\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=LSB ALIGN=MIDDLE  ALT=\"LSB\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=LBO ALIGN=MIDDLE  ALT=\"LBO\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=SB%2B+BO=8.2m ALIGN=MIDDLE  ALT=\"SB%2B+BO=8.2m\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%2Atan%2828%5Eo%29%2BL%2Atan%2835%5Eo%29=8.2m ALIGN=MIDDLE  ALT=\"L%2Atan%2828%5Eo%29%2BL%2Atan%2835%5Eo%29=8.2m\"   DISABLED_style= \"max-width:90%; height:auto;\" >   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%28tan%2828%5Eo%29%2Btan%2835%5Eo%29%29=8.2m ALIGN=MIDDLE  ALT=\"L%28tan%2828%5Eo%29%2Btan%2835%5Eo%29%29=8.2m\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L=8.2m%2F1.231917=6.656293m ALIGN=MIDDLE  ALT=\"L=8.2m%2F1.231917=6.656293m\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance between Lina and the Orb is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%2Fcos%2835%5Eo%29=6.656293m%2F0.819152=highlight%288.126m%29 ALIGN=MIDDLE  ALT=\"L%2Fcos%2835%5Eo%29=6.656293m%2F0.819152=highlight%288.126m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance between Kai and the Orb is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%2Fcos%2842%5Eo%29=6.656293m%2F0.743145=highlight%288.957m%29 ALIGN=MIDDLE  ALT=\"L%2Fcos%2842%5Eo%29=6.656293m%2F0.743145=highlight%288.957m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If Orin magically lifts the Silver Orb 1.5 meters higher, it will be at point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22S%27%22 ALIGN=MIDDLE  ALT=\"%22S%27%22\"   DISABLED_style= \"max-width:90%; height:auto;\" >, 1.5 meters above <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S ALIGN=MIDDLE  ALT=\"S\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "In triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=KAS ALIGN=MIDDLE  ALT=\"KAS\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AS=L%2Atan%2842%5Eo%29=6.656293m%2A0.900404=5.993353m ALIGN=MIDDLE  ALT=\"AS=L%2Atan%2842%5Eo%29=6.656293m%2A0.900404=5.993353m\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "In the new right triangle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22KAS%27%22 ALIGN=MIDDLE  ALT=\"%22KAS%27%22\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22AS%27%22=5.993353m-1.5m=4.493353m ALIGN=MIDDLE  ALT=\"%22AS%27%22=5.993353m-1.5m=4.493353m\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The length of side <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=KA ALIGN=MIDDLE  ALT=\"KA\"   DISABLED_style= \"max-width:90%; height:auto;\" > is still <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6.656293m ALIGN=MIDDLE  ALT=\"6.656293m\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Kai's distance to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22S%27%22 ALIGN=MIDDLE  ALT=\"%22S%27%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > in meters can be calculated from the Pythagorean theorem as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22KS%27%22=sqrt%286.65629%5E2%2B4.493353%5E2%29=8.030968413 ALIGN=MIDDLE  ALT=\"%22KS%27%22=sqrt%286.65629%5E2%2B4.493353%5E2%29=8.030968413\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%288.031m%29 ALIGN=MIDDLE  ALT=\"highlight%288.031m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > when rounded to 3 decimal places. That puts Kai closer to the Silver Orb than Lina.<BR>\n" );
document.write( "Orin could instantly calculate all that. His magical powers make him a geometry and trigonometry wizard too. \r<BR>\n" );
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