SOLUTION: If there is anyone who knows how to do this I would appreciate it if you can help me. This is due tommorrow and I am not sure where to even begin!! Here is a question I really am

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: If there is anyone who knows how to do this I would appreciate it if you can help me. This is due tommorrow and I am not sure where to even begin!! Here is a question I really am       Log On


   

Question 129051: If there is anyone who knows how to do this I would appreciate it if you can help me. This is due tommorrow and I am not sure where to even begin!!
Here is a question I really am lost on! I don't know what to do and am just confusing myself more. Can someone Please HelP??? In the early 1990's Iomega came out with a portable disk storage device that was capable of storing 100mb of data. There was a huge market and absolutely no competitors. For several months Iomega experienced exponential growth. If we modeled that growth using the exponential equation Growth=.05e^t where t is the number of months, then the company would double its value in 3.7 months. Unfortunately for Iomega and their shareholders, they did not realize that such growth would not continue for long. In fact, competitors began releasing their products very quickly and Iomega began experiencing logistic growth rather than exponential growth. In fact, their growth would be better modeled by the equation G=100/1+1000e^-t . This is still a nice growth, but it says that they would eventually maximize their growth at 100 times their initial size. The two groth functions intersect. That is, at some number of months .05t=G = 100/1+1000e^-t. After this time, their growth would be slower than the exponential model would predict, and eventually they would level out at a very nearly no growth. A. Your job is to find that point where the two functions intersect. You will need to use the natural log and simple algebra to solve this equation. B. How many times larger than their initial value would they be at this point? HELP!!!!!!!!!
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
This problem is just loaded with lots of extra information that you do not need. You just have
to sort through it and disregard the extra stuff. Without a lot of explanation, here's what you
end up doing:
.
To find the common solution of the two problems, set the right sides of the two Growth equations
equal as follows:
.
0.05e%5Et+=+100%2F%281%2B1000e%5E%28-t%29%29
.
Divide both sides by 100 to reduce the numerator on the right side and get:
.
0.0005e%5Et+=+1%2F%281%2B1000e%5E%28-t%29%29
.
Get rid of the denominator on the right side by multiplying both sides by 1%2B1000e%5E%28-t%29.
When you multiply both sides by this quantity it cancels the denominator on the right side
and the equation becomes:
.
%280.0005e%5Et%29%2A%281%2B1000e%5E%28-t%29%29=+1
.
Multiply out the left side. Note that when you multiply e%5Et%2Ae%5E%28-t%29 you add the exponents
to get e%5E0 and any quantity raised to the zero power is 1. So the distributed
multiplication on the left side results in:
.
0.0005e%5Et+%2B+0.5+=+1
.
Subtract the 0.5 from both sides and you have:
.
0.0005e%5Et+=+0.5
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Divide both sides by 0.0005 and you get:
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e%5Et+=+0.5%2F0.0005+=+1000
.
Now take the natural log of both sides to get:
.
ln%28e%5Et%29+=+ln%281000%29
.
On the left side apply the exponent rule of logarithms and you have:
.
t%2Aln%28e%29+=+ln%281000%29
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Using a calculator you can find that ln%28e%29+=+1 and ln%281000%29=+6.907755279.
Substitute these values and the equation reduces to:
.
t+=+6.907755279
.
This tells you that when t equals 6.907755279 months the growth changes over from exponential
growth to logistic growth.
.
To find the growth at that point in time, you can substitute this value of t into either
of the two growth equations you were originally given. No sense in making it any harder than
it needs to be. Let's just substitute that value of t into the equation G+=+0.05e%5Et. When
you substitute 6.907755279 for t this equation becomes:
.
G+=+0.05e%5E6.907755279
.
Using a calculator you can find that e%5E6.907755279+=+1000. This makes the equation become:
.
G+=+0.05%2A1000+=+50
.
This tells you at the point where the two graphs intersect the value of t is 6.907755279 months
and the value of the Growth at that point is 50 times its original value. So the answer to
Part B of this problem is that the growth is 50 times larger than its original value.
.
This is confirmed by the fact that you are told that when t = 3.7 months the growth is given
as 2 times its original value and you can calculate that from using t = 3.7 and the equation
.
G+=+0.05e%5E3.7+=+0.05%2A40.44730435+=+2.022365218
.
And this is approximately the 2 that the problem says it should be.
.
Hope this gives you an insight into the problem and how to solve it.
.