document.write( "Question 1202012: If 80​% of a radioactive element remains radioactive after 200 million​ years, then what percent remains radioactive after 700 million​ years? What is the half life of this element? \n" ); document.write( "
Algebra.Com's Answer #836628 by Theo(13342)\"\" \"About 
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formua you can use is f = p * (1+r) ^ n
\n" ); document.write( "f is the future value
\n" ); document.write( "p is the present value
\n" ); document.write( "r is the growth rate per time period.
\n" ); document.write( "2 + r is the growh factor per time period.
\n" ); document.write( "n is the number of time periods\r
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\n" ); document.write( "\n" ); document.write( "the time periods are in millions of years.\r
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\n" ); document.write( "\n" ); document.write( "after 200 million years, 80% of the element remains radioactive.
\n" ); document.write( "formula becomes:
\n" ); document.write( ".8 = 1 * (1 + r) ^ 200
\n" ); document.write( "divide both sides of the equation by 1 to get:
\n" ); document.write( ".8/1 = (1 + r) ^ 200
\n" ); document.write( "solve for (1 + r) to get:
\n" ); document.write( "(1 + r) = (.8/1) ^ (1 / 200) = .9988849044\r
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\n" ); document.write( "\n" ); document.write( "that says that the growth factor is .9988849044 every million years.\r
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\n" ); document.write( "\n" ); document.write( "to confirm, replace n in the original equation and solve for f to get:
\n" ); document.write( "f = 1 * .9988849044 ^ 200 = .8\r
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\n" ); document.write( "\n" ); document.write( "now that you have the growth factor for every 1 million years, you can solve for the remaining percent after 700 million years.\r
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\n" ); document.write( "\n" ); document.write( "the formula becomes f = 1 * .9988849044 ^ 700 = .4579467218 = 45/80%.\r
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\n" ); document.write( "\n" ); document.write( "to find the half life, set f = .5 in the original equation and solve for n.\r
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\n" ); document.write( "\n" ); document.write( "you will get:
\n" ); document.write( ".5 = 1 * .9988849044 ^ n
\n" ); document.write( "divide both sides of the eqution by 1 to get:
\n" ); document.write( ".5/1 = .9988849044 ^ n
\n" ); document.write( "simplify to get:
\n" ); document.write( ".5 = .9988849044 ^ n
\n" ); document.write( "take the log of both sides of the equation to get:
\n" ); document.write( "log(.5) = log(.9988849044 ^ n)
\n" ); document.write( "by log rule that says log(x^n) = n * log(x), this becomes:
\n" ); document.write( "log(.5) = n * log(.9988849044)
\n" ); document.write( "divide both sides of this equation by log(.9988849044) to get:
\n" ); document.write( "log(.5) / log(.9988849044) = n
\n" ); document.write( "solve for n to get:
\n" ); document.write( "n = 621.2567439 million years.\r
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\n" ); document.write( "\n" ); document.write( "to confirm, replace n in the original eqution and solve for f to get:
\n" ); document.write( "f = 1 * .9988849044 ^ 621.2567439 = .5\r
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\n" ); document.write( "\n" ); document.write( "the equation can be graphed as shown below:\r
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