SOLUTION: Use the sales decline function {{{S(t)=Se^(-at)}}}. If a=1, S=50,000, and t is time measured in years, find the number of years it will take for sales to fall to half the initial

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Question 44912: Use the sales decline function S%28t%29=Se%5E%28-at%29. If a=1, S=50,000, and t is time measured in years, find the number of years it will take for sales to fall to half the initial sales.
Answer by Chris435435(8) About Me  (Show Source):
You can put this solution on YOUR website!
For this problem, they give you a=1 and S=50000. So plug those into the formula and you get:
S%28t%29=+50000e%5E%28-t%29
Since they want us to find the number of years it will take for sales to fall to half the initial sales, we must compute the initial sales. So, we must set t = 0 since this is the the beginning of our sales, our initial sales. So we get
S%28t%29+=+50000e%5E%280%29+=+50000
Since S(t) = 50000, we must take half of this amount and find the number of years when the sales will decline to this halved amount which is 50000%2F2=25000. So our equation would now look like:
25000+=+50000e%5E%28-t%29
So solving for t:
0.5+=+e%5E%28-t%29
ln%280.5%29+=+-t
-ln%280.5%29+=+t
t+=+0.693
So, it'll take approximately 0.693 years for sales to decline to this halved amount.