Question

Two years ago, a plant was $192$ cm tall. At present its height is $243$ cm. Find the rate of its growth.

Answer 1

Hint: In this question we have been given the present height of a plant which is given to us as $243$ cm and the height of the plant $2$ years ago was $192$ cm. We have to find the rate of growth of the plant in $2$ years. We will use the formula of compound interest and substitute the values for the present amount and the past amount and the number of years in the formula which is $A=P{{\left( 1+\dfrac{r}{100} \right)}^{T}}$ where, $A$ is the final height, $P$ is the initial height, $r$ is the rate of growth and $T$ is the number of years.

Complete step by step solution:
We have from the question the initial height $P=192$ cm, the final height $A=243$ cm and the number of years $T=2$.
On substituting these values in the formula, we get:
$\Rightarrow 243=192{{\left( 1+\dfrac{r}{100} \right)}^{2}}$
On transferring $192$ from the right-hand side to the left-hand side, we get:
$\Rightarrow \dfrac{243}{192}={{\left( 1+\dfrac{r}{100} \right)}^{2}}$
We can simplify the left-hand side, we get:
$\Rightarrow \dfrac{81}{64}={{\left( 1+\dfrac{r}{100} \right)}^{2}}$
On taking the square root on both the sides, we get:
$\Rightarrow \sqrt{\dfrac{81}{64}}=\left( 1+\dfrac{r}{100} \right)$
Now we know the square root of $81$ is $9$ and the square root of $64$ is $8$ therefore, we can write it as:
$\Rightarrow \dfrac{9}{8}=\left( 1+\dfrac{r}{100} \right)$
On taking the lowest common multiple on the right-hand side, we get:
$\Rightarrow \dfrac{9}{8}=\dfrac{100+r}{100}$
On cross multiplying, we get:
$\Rightarrow 9\times 100=8\times \left( 100+r \right)$
On simplifying, we get:
$\Rightarrow 900=800+8r$
On transferring $800$ from the right-hand side to the left-hand side, we get:
$\Rightarrow 900-800=8r$
On simplifying and rearranging the terms, we get:
$\Rightarrow r=\dfrac{100}{8}$
On dividing, we get:
$\Rightarrow r=12.5\%$, which is the required answer.

Note: It is to be remembered that the formula we used is for compound interest value of the growth. Because the growth of a plant each year depends on the height of it in the previous year. In this question when we took the square root, we neglected the negative values of the root because that would give us a negative rate of growth which means that the height is decreasing therefore, we only consider the positive values.