Question

The number of trees required to plant in an estate varies inversely as square of the distance between trees. If $500$ trees are required when the distance between them is $3.3m$, how many trees are required when the distance between the plants is $5.5m$? What is the distance between the trees when $165$ trees are planted?

Answer 1

If $t$ is the number of trees to be planted and $d$ is the distance between the trees, then
$t\propto \frac{1}{d^2}$ or $t{d}^2=k$,
for some constant $k$. If $t_1$ corresponds to $d_1$ and $t_2$ corresponds to $d_2$, we obtain
$t_1{d}_1^2=t_2{d}_2^2$.
We know $t_1=500$ when $d_1=3.3$. We have to find $t_2$ when $d_2=5.5$. Hence
$t_2=\frac{t_1{d}_1^2}{d_2^2}=\frac{500\times (3.3{)}^2}{(5.5{)}^2}=\frac{500\times 9}{25}=180$.
Thus $180$ trees can be planted when the distance between the trees is $5.5m$.
If $t_2=165$, what is $d_2$? The same relation gives
$d_2^2=\frac{t_1{d}_1^2}{t_2}=\frac{500\times (5.5{)}^2}{165}=33$.
Hence
$d_2=\sqrt{33}\approx 5.745m$.