document.write( "Question 1130: If 23.00 grams of a 45.00 gram sample of unknown isotope remain after 17 days 12 hours, what is the half-life of the unknown isotope to the nearest minute? \n" ); document.write( "

Algebra.Com's Answer #338 by AnlytcPhil(1806) $\"\"$ $\"About$

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If 23.00 grams of a 45.00 gram sample of unknown isotope remain after 
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document.write( "17 days 12 hours, what is the half-life of the unknown isotope to the 
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document.write( "nearest minute?
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document.write( "The formula is 
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document.write( "               A = Pe^rt
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document.write( "When time t=0 days, the amount A = 45g.  Substitute these two values
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document.write( "              45 = P^er(0)
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document.write( "              45 = Pe⁰
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document.write( "              45 = P(1)
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document.write( "               P = 45 grams
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document.write( "so substitute 45 for P in A = Pe^rt
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document.write( "               A = 45ert
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document.write( "When time t=17.5 days, the amount A = 23g.  Substitute these two values:
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document.write( "              23 = 45e^r(17.5)
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document.write( "              23 = 45e^17.5r
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document.write( "Divide both sides by 45
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document.write( "           23/45 = e^17.5r
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document.write( "     .5111111111 = e^17.5r
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document.write( "Rewrite this equation using the fact that
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document.write( "\" N = e^M can be rewritten as M = ln(N) \"
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document.write( "           17.5r = ln(.5111111111)
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document.write( "Use a calculator to find ln(.5111111111) = -.6711682738
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document.write( "           17.5r = -.6711682738
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document.write( "Solve for r
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document.write( "               r = -.6711682738/17.5
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document.write( "               r = -.0383524728
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document.write( "So substitute this value of r in A = 45e^rt
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document.write( "               A = 45e^{-.0383524728t}
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document.write( "We wish to know how many minutes is required for it to become half
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document.write( "its original amount, or 1/2 of 45g, or 22.5 g.  So substitute this 
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document.write( "for A in A = 45e^{-.0394804867t}
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document.write( "            22.5 = 45e^{-.0383524728t}
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document.write( "Divide both sides by 45
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document.write( "         22.5/45 = e^{-.0383524728t}
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document.write( "              .5 = e^{-.0383524728t}
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document.write( "Rewrite this as 
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document.write( "          ln(.5) = -.0383524728t
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document.write( "Solve for t
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document.write( "    -.6931471806 = -.0383524728t
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document.write( "   -.0383524728t = -.6931471806
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document.write( "               t = -.6931471806/(-.0383524728)
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document.write( "               t = 18.07307665 days
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document.write( "To find hours, multiply .07307665 by 24
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document.write( "       18 days 1.753839658 hours
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document.write( "To find minutes, multiply .753839658 by 60
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document.write( "       18 days 1 hour 45.23037947 minutes or to nearest minute:
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document.write( "       18 days 1 hour, 45 minutes.
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document.write( "Edwin\r
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