document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1032718>Question 1032718</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A satellite has a radioactive isotope power supply. The power output in watts is given by the equation P = 50e^(t/250), where P is the power in watts and t is the time in days. \r<BR>\n" );
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document.write( "a ) What is the half life of the power supply?\r<BR>\n" );
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document.write( "b) The equipment aboard the satellite requires 10 watts of power to operate properly. What is the operational life of the satellite?\r<BR>\n" );
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document.write( "I am  having trouble setting up these equations properly. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #647328 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=647328\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Question: <BR>\n" );
document.write( "A satellite has a radioactive isotope power supply. The power output in watts is given by the equation P = 50e^(t/250), where P is the power in watts and t is the time in days. <BR>\n" );
document.write( "a ) What is the half life of the power supply? <BR>\n" );
document.write( "b) The equipment aboard the satellite requires 10 watts of power to operate properly. What is the operational life of the satellite?\r<BR>\n" );
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document.write( "Since this is decaying, the exponent needs to have a negative in it. So it should be<BR>\n" );
document.write( "P = 50e^(-t/250)\r<BR>\n" );
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document.write( "Part (a)  What is the half life of the power supply?\r<BR>\n" );
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document.write( "Plug in t = 0 to get...<BR>\n" );
document.write( "P = 50e^(-t/250)<BR>\n" );
document.write( "P = 50e^(0/250)<BR>\n" );
document.write( "P = 50e^(0)<BR>\n" );
document.write( "P = 50*1<BR>\n" );
document.write( "P = 50<BR>\n" );
document.write( "So at t = 0 days (aka the beginning), the power output in watts is 50 watts\r<BR>\n" );
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document.write( "So we start off with 50 watts. The question is: how long will it take to get to 25 watts? Which is half of the original output. \r<BR>\n" );
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document.write( "Let's find out. Plug in P = 25 and solve for t.\r<BR>\n" );
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document.write( "P = 50e^(-t/250)<BR>\n" );
document.write( "25 = 50e^(-t/250)<BR>\n" );
document.write( "25/50 = [50e^(-t/250)]/50<BR>\n" );
document.write( "0.5 = e^(-t/250)<BR>\n" );
document.write( "e^(-t/250) = 0.5<BR>\n" );
document.write( "Ln[e^(-t/250)] = Ln[0.5]<BR>\n" );
document.write( "(-t/250)*Ln[e] = Ln[0.5]<BR>\n" );
document.write( "(-t/250)*1 = Ln[0.5]<BR>\n" );
document.write( "-t/250 = Ln[0.5]<BR>\n" );
document.write( "t = -250*Ln[0.5]<BR>\n" );
document.write( "t = 173.286795\r<BR>\n" );
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document.write( "It takes approximately <font color=red>173.286795 days</font> for the amount of 50 watts to turn into 25 watts.<BR>\n" );
document.write( "So the half-life is approximately <font color=red>173.286795 days</font>\r<BR>\n" );
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document.write( "Part (b) The equipment aboard the satellite requires 10 watts of power to operate properly. What is the operational life of the satellite?\r<BR>\n" );
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document.write( "Now we need to find out how long this power supply will work as long as P >= 10. Let's plug in P = 10, then solve for t, to find out\r<BR>\n" );
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document.write( "P = 50e^(-t/250)<BR>\n" );
document.write( "10 = 50e^(-t/250)<BR>\n" );
document.write( "10/50 = [50e^(-t/250)]/50<BR>\n" );
document.write( "0.2 = e^(-t/250)<BR>\n" );
document.write( "e^(-t/250) = 0.2<BR>\n" );
document.write( "Ln[e^(-t/250)] = Ln[0.2]<BR>\n" );
document.write( "(-t/250)*Ln[e] = Ln[0.2]<BR>\n" );
document.write( "(-t/250)*1 = Ln[0.2]<BR>\n" );
document.write( "-t/250 = Ln[0.2]<BR>\n" );
document.write( "t = -250*Ln[0.2]<BR>\n" );
document.write( "t = 402.359478\r<BR>\n" );
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document.write( "So the power supply will last for approximately <font color=red>402.359478 days</font> til it reaches 10 watts. After that, it won't operate properly since it won't produce enough watts. \r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "Answer to part (a): <font color=red>173.286795 days</font> (this is approximate)<BR>\n" );
document.write( "Answer to part (b): <font color=red>402.359478 days</font> (this is approximate) </SPAN>\n" );
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