Question 129051
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:
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To find the common solution of the two problems, set the right sides of the two Growth equations
equal as follows:
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{{{0.05e^t = 100/(1+1000e^(-t))}}}
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Divide both sides by 100 to reduce the numerator on the right side and get:
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{{{0.0005e^t = 1/(1+1000e^(-t))}}}
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Get rid of the denominator on the right side by multiplying both sides by {{{1+1000e^(-t)}}}. 
When you multiply both sides by this quantity it cancels the denominator on the right side
and the equation becomes:
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{{{(0.0005e^t)*(1+1000e^(-t))= 1}}}
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Multiply out the left side. Note that when you multiply {{{e^t*e^(-t)}}} you add the exponents
to get {{{e^0}}} and any quantity raised to the zero power is 1. So the distributed
multiplication on the left side results in:
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{{{0.0005e^t + 0.5 = 1}}}
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Subtract the 0.5 from both sides and you have:
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{{{0.0005e^t = 0.5}}}
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Divide both sides by {{{0.0005}}} and you get:
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{{{e^t = 0.5/0.0005 = 1000}}}
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Now take the natural log of both sides to get:
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{{{ln(e^t) = ln(1000)}}}
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On the left side apply the exponent rule of logarithms and you have:
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{{{t*ln(e) = ln(1000)}}}
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Using a calculator you can find that {{{ln(e) = 1}}} and {{{ln(1000)= 6.907755279}}}.
Substitute these values and the equation reduces to:
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{{{t = 6.907755279}}}
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This tells you that when t equals 6.907755279 months the growth changes over from exponential
growth to logistic growth.
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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^t}}}. When 
you substitute 6.907755279 for t this equation becomes:
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{{{G = 0.05e^6.907755279}}}
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Using a calculator you can find that {{{e^6.907755279 = 1000}}}. This makes the equation become:
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{{{G = 0.05*1000 = 50}}}
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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.
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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
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{{{G = 0.05e^3.7 = 0.05*40.44730435 = 2.022365218}}}
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And this is approximately the 2 that the problem says it should be.
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Hope this gives you an insight into the problem and how to solve it. 
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