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An elephant weighs 120 kg after 7 days , after 30 days it weights 150kg, how much does the elephant weights after 150 days after it's born? When it's always exponential

Looks like we want to use an exponential to model the elephant's weight gain. This can be written as

W(t) = B･exp(at)

where t is time (in days), A = growth constant and B = initial weight of elephant (when it was born; i.e., weight W(0) = B when t = 0).

We have 2 'data points' to use to determine the 2 parameters A and B. These are

W(7) = 120 = B･exp(7A)

W(30) = 150 = B･exp(30A).

Now its just a matter of a little algebra to find A and B. It looks like we can make some progress by taking the (natural) log of the 2 equations to get (after dividing by B)

ln(120/B) = 7A --> ln(120) - ln(B) = 7A (using the property of logs that ln(x/y) = ln(x) - ln(y) ) --> ln(B) = ln(120) - 7A

ln(150/B) = 30a --> ln(150) - ln(B) = 30A or ln(B) = ln(150) - 30A

Setting the 2 expressions for ln(B) equal to each other gives

ln(120) - 7A = ln(150) - 30A --> a few algebra steps --> ln(150/120)/23 = A --> a ≈ 0.01.

So now we know the parameter A. To get B, plug this value into one of the previous expressions for ln(B) and exponentiate to get

B = exp{ln(120) - 7A) = 120･exp(-0.07) ≈ 112.

With the 2 parameters determined, we can evaluate

W(150) = 112･exp{(0.01)･150) = 502 kg.

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