A tree if growing taller at continuous rate. In past 12 years it has grown from 3 meters to 15 meters.

Question

I have to find the growth percentage of tree per year

I have applied this technique : $12yr=3\times e^{(r\times 12)}$

where $r$ is the rate which I have to find

But nothing happens this way .

Can someone help me solve this question?

Answer 1

Let the rate per annum be $r.$ Then we have that after the first year, the height of the tree is $3(1+r).$ After the second year that's $3(1+r)^2,$ and after the twelfth year the tree is $$3(1+r)^{12}=15.$$ You are to solve this for a positive value of $r.$ Note that this is equivalent to $(1+r)^{12}=5,$ so that $1+r=5^{1/12},$ or that $$r=5^{1/12}-1,$$ which is positive provided we take the positive root out of the twelve. You may approximate this with a calculator.

Answer 2

This can be solved by taking log on both sides.

Growth formula is given by $$ P = P_0 e^{rt} $$

Here, $P_0$ and $P$ are initial and final length of the tree, respectively, $r$ is continuously compounded rate of growth, $t$ is time in years.

$$15 = 3 e^{r \times 12}$$ $$\log_e(15 / 3) = r \times 12 $$

$$r \approx 0.1341$$

Hence the growth rate is 13.41% per year.