Question 159133
P = percentage = 95
13+14*ln(x) = 95
ln(x) = (95-13)/14
ln(x) = 5.857142857
{{{y = ln(x)}}} if and only if {{{e^y = x}}}
y = ln(x) = 5.857142857, so y = 5.857142857
equation for {{{e^y = x}}} becomes {{{e^(5.857142857) = x}}}
use the calculator to solve for {{{e^(5.857142857)}}}
answer becomes x = 349.7235051
substituting in the original equation of
{{{95 = 13 + (14 * ln(x))}}}, we get
{{{95 = 13 + (14 * ln(349.7235051))}}}
solving for ln(349.7235051) using the calculator, we get
{{{95 = 13 + (14 * 5.857142857)}}}, which becomes
{{{95 = 13 + 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(1200,1200,-100,400,-50,100,(14*ln(x))+13)}}}
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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*ln(x)+13}}} 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.