SOLUTION: Solve the problem. Please show your work. 1. The population of a colony of bacteria is decaying exponentially according to the function below, where t is the time in minutes. Ho

Algebra ->  Rational-functions -> SOLUTION: Solve the problem. Please show your work. 1. The population of a colony of bacteria is decaying exponentially according to the function below, where t is the time in minutes. Ho      Log On


   

Question 214806: Solve the problem. Please show your work.
1. The population of a colony of bacteria is decaying exponentially according to the function below, where t is the time in minutes. How many minutes will it take for the population of the colony to drop to 50?
B(t)=2000*e^(-.03*t)
e=2.718
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
B%28t%29=2000%2Ae%5E%28-.03%2At%29
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Substitute 50 for B(t) to get:
50+=+2000%2Ae%5E%28-.03%2At%29
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Take the natural log of both sides to get:
ln%2850%29+=+ln%282000%2Ae%5E%28-.03%2At%29%29
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rules of logarithms apply here.
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first rule:
ln%28a%2Ab%29+=+ln%28a%29+%2B+ln%28b%29
second rule:
ln%28a%5Eb%29+=+b%2Aln%28a%29
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applying the first rule,
ln%282000%2Ae%5E%28-.03%2At%29%29 becomes ln%282000%29+%2B+ln%28e%5E%28-.03t%29%29
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applying the second rule,
ln%28e%5E%28-.03t%29%29 becomes -.03t%2Aln%28e%29
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putting this all together, your equation becomes:
ln%2850%29+=+ln%282000%29+-+.03t%2Aln%28e%29
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ln%28e%29 equals 1, so your equation becomes:
ln%2850%29+=+ln%282000%29+-+.03t
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subtract ln(2000) from both sides of this equation and then divide both sides of this equation by (-.03) to get:
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%28ln%2850%29+-+ln%282000%29%29%2F%28-.03%29+=+t
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this becomes:
%283.912023005+-+7.60090246%29%2F%28-.03%29+=+t
this becomes:
122.9626485+=+t
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The population will decay to 50 in 122.9626485 hours.
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substitute this value for t in the original equation to get:
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B%28t%29=2000%2Ae%5E%28-.03%2A%28122.9626485%29%29
this becomes:
B%28t%29+=+50 confirming that t+=+122.9626485 is a good value for t.
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