Answer: 12 months
Step-by-step explanation:
Given : A tree double in weight in three months.
Since the weight is increasing by growth factor of 2 , therefore its is an exponential growth.
The exponential growth equation is given by :-
(1)
, where A is the initial values , b is the growth factor and x is the time period.
As per given , b= 2
Since the tree doubles in weight in three months, so time period x =
, where t= number of months.
Substitute the value of b and x in (1) , we get
, where y= weight of tree after t months and A is initial weight of tree.
When it will be 1600% of his initial weight , the weight of tree : y= 1600% of A =![\dfrac{1600}{100}\times A=16A](https://tex.z-dn.net/?f=%5Cdfrac%7B1600%7D%7B100%7D%5Ctimes%20A%3D16A)
At y= 16 A , ![16A=A(2)^{\dfrac{t}{3}}](https://tex.z-dn.net/?f=16A%3DA%282%29%5E%7B%5Cdfrac%7Bt%7D%7B3%7D%7D)
![\Rightarrow\ 16=2^{\dfrac{t}{3}}](https://tex.z-dn.net/?f=%5CRightarrow%5C%2016%3D2%5E%7B%5Cdfrac%7Bt%7D%7B3%7D%7D)
![\Rightarrow\ 2^{4}=2^{\dfrac{t}{3}}](https://tex.z-dn.net/?f=%5CRightarrow%5C%202%5E%7B4%7D%3D2%5E%7B%5Cdfrac%7Bt%7D%7B3%7D%7D)
![\Rightarrow\ 4=\dfrac{t}{3}\Rightarrow\ t=3\times4=12](https://tex.z-dn.net/?f=%5CRightarrow%5C%204%3D%5Cdfrac%7Bt%7D%7B3%7D%5CRightarrow%5C%20t%3D3%5Ctimes4%3D12)
Hence, it will take 12 months to be 1600% in weight.