Question

In a forest there are 40,000 trees. Find the expected number of trees after 3 years if the objective is to increase the number at the rate 5% per year.

Answer 1

Hint:The number of trees at the start of \[{{1}^{st}}\] year is 40,000. The trees are increasing at the rate of 5%. Now, calculate the number of trees at the end of \[{{1}^{st}}\] year is \[40000+5\%\,of\,40000\] . The number of trees at the start of \[{{2}^{nd}}\] year is equal to the number of trees at the end of \[{{1}^{st}}\] year. The trees are increasing at the rate of 5%. Now, calculate the number of trees at the end of \[{{2}^{nd}}\] year is \[42000+5\%\,of\,42000\] . The number of trees at the start of \[{{3}^{rd}}\] year is equal to the number of trees at the end of \[{{2}^{nd}}\] year. Now, calculate the trees are increasing at the rate of 5%. The number of trees at the end of \[{{3}^{rd}}\] year is \[44100+5\%\,of\,44100\] . Now, calculate it further and get the answer.

Complete step-by-step answer:
According to the question, it is given that in a forest there are 40,000 trees and the number is increasing at the rate of 5% per year.
The total number of trees in the forest = 40,000 ……………..(1)
The rate by which the trees are increasing is 5% and we have to calculate the expected number of trees after 3 years.
Let us calculate the total number of trees every year.
First of all, calculating the number of trees at the end of \[{{1}^{st}}\] year.
The number of trees at the start of \[{{1}^{st}}\] year = 40,000 ………………(2)
Rate of increasing tree = 5% ……………(3)
The increased number of trees at the end of \[{{1}^{st}}\] year = \[40000+5\%\,of\,40000\]
\[\begin{align}
  & =40000+\left( \dfrac{5}{100}\times 40000 \right) \\
 & =40000+2000 \\
 & =42000 \\
\end{align}\]
So, the number of trees at the end of \[{{1}^{st}}\] year is 42,000.
Now, calculating the number of trees at the end of \[{{2}^{nd}}\] year.
The number of trees at the start of \[{{2}^{nd}}\] year = 42,000.
Rate of increasing tree = 5%.
The increased number of trees at the end of \[{{2}^{nd}}\] year = \[42000+5\%\,of\,42000\]
\[\begin{align}
  & =42000+\left( \dfrac{5}{100}\times 42000 \right) \\
 & =42000+2100 \\
 & =44100 \\
\end{align}\]
So, the number of trees at the end of \[{{2}^{nd}}\] year is 44,100.
Now, calculating the number of trees at the end of \[{{3}^{rd}}\] year.
The number of trees at the start of \[{{3}^{rd}}\] year = 44,100.
Rate of increasing tree = 5%.
The increased number of trees at the end of \[{{3}^{rd}}\] year = \[44100+5\%\,of\,44100\]
\[\begin{align}
  & =44100+\left( \dfrac{5}{100}\times 441000 \right) \\
 & =44100+2205 \\
 & =46305 \\
\end{align}\]
So, the number of trees at the end of \[{{3}^{rd}}\] year is 46,305.
Hence, the expected number of trees after 3 years if the trees are increasing at the rate 5% per year is 46,305.

Note: We can also solve this question by direct formula,
\[\text{Final Number of trees =Initial number of trees}{{\left( 1+\dfrac{Rate}{100} \right)}^{n}}\] .
Here, n is the number of years.
Initial number of trees = 42000.
Rate =5%
Number of years, n=3 years
Now, using the formula, we get
\[\begin{align}
  & \text{Final Number of trees =40000}{{\left( 1+\dfrac{5}{100} \right)}^{3}} \\
 & =40000\times \dfrac{105}{100}\times \dfrac{105}{100}\times \dfrac{105}{100} \\
 & =4\times 105\times 105\times \dfrac{105}{100} \\
 & =4\times 105\times 105\times \dfrac{21}{20} \\
\end{align}\]
\[\begin{align}
  & =4\times 105\times 21\times \dfrac{21}{4} \\
 & =105\times 21\times 21 \\
 & =46305 \\
\end{align}\]
Hence, the expected number of trees after 3 years if the trees are increasing at the rate 5% per year is 46,305.