SOLUTION: Suppose that the function P = 13 + 14 ln x, represents the perecentage of inbound e-mail in the US. that is considered Spam, where x represents the number of years after 2000. Use

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Suppose that the function P = 13 + 14 ln x, represents the perecentage of inbound e-mail in the US. that is considered Spam, where x represents the number of years after 2000. Use      Log On


   

Question 159133: Suppose that the function P = 13 + 14 ln x, represents the perecentage of inbound e-mail in the US. that is considered Spam, where x represents the number of years after 2000.
Use this model to determine in how many years(to two decimal places) it will take for the percentage to reach 95%, provided that law enforcement regarding spammers does not change.
Answer by gonzo(654) About Me  (Show Source):
You can put this solution on YOUR website!
P = percentage = 95
13+14*ln(x) = 95
ln(x) = (95-13)/14
ln(x) = 5.857142857
y+=+ln%28x%29 if and only if e%5Ey+=+x
y = ln(x) = 5.857142857, so y = 5.857142857
equation for e%5Ey+=+x becomes e%5E%285.857142857%29+=+x
use the calculator to solve for e%5E%285.857142857%29
answer becomes x = 349.7235051
substituting in the original equation of
95+=+13+%2B+%2814+%2A+ln%28x%29%29, we get
95+=+13+%2B+%2814+%2A+ln%28349.7235051%29%29
solving for ln(349.7235051) using the calculator, we get
95+=+13+%2B+%2814+%2A+5.857142857%29, which becomes
95+=+13+%2B+82, which becomes
95 = 95.
answer is 349.72 years rounding to the nearest hundredth.
now that we know how far out x has to go to satisfy the equation, we can graph it as follows:
range of x = -1 to 400 where x represents years from 2000.
range of y = -10 to 100 where y represents percent.
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graph looks like this
please scan below the graph for further comments.
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graph%281200%2C1200%2C-100%2C400%2C-50%2C100%2C%2814%2Aln%28x%29%29%2B13%29
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range of x had to start from - something in order for the x-axis to display properly.
range of y had to start from - something in order for the y-axis to display properly
the formula of y+=+14%2Aln%28x%29%2B13 is not defined for x <= 0 because e^y will never be negative and e^y can approach 0 but never be 0.
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based on the formula as shown, x = 349.72 rounded to the nearest hundredth of a year.
that year will be 2000 + 349.72 = 2349.72 based on the formula provided.