Question

Mr. Dinesh owns an rectangular agricultural farm at village Talvel. The length of the farm is 10 metre more than twice the breadth. In order to harvest rain water, he dug a square shaped pond inside the farm. The side of pond is $\frac{1}{3}$ of the breadth of the farm. The area of the farm is 20 times the area of the pond. Find the length and breadth of the farm and side of the pond.

Answer 1

Let the breadth of the rectangular farm be $xm$.
$\therefore$ Length of rectangular farm $=(2x+10)m$
Area of rectangular farm $=$ Length $\times$ Breadth
$\begin{array}{cc}& =(2x+10)\times x\\ & =(2x^2+10x)\text{sq.}m\end{array}$
Now, side of square shaped pond $=\frac{x}{3}m$
$\therefore$ Area of square shaped pond $=(\text{side}{)}^2$
$\begin{array}{cc}& ={\left(\frac{x}{3}\right)}^{2}m\\ & =\frac{x^2}{9}m\end{array}$
According to the given condition,
Area of rectangular farm $=20\times$ Area of pond
$\begin{array}{cc}\therefore & 2x^2+10x=20\times \left(\frac{x^2}{9}\right)\\ \therefore & x^2+5x=\frac{10x^2}{9}\dots [\text{Dividing both sides by}2\text{]}\\ \therefore & 9x^2+45x=10x^2\dots [\text{Multiplying both sides by}9\text{]}\\ & \therefore 10x^2-9x^2-45x=0\\ \therefore & x^2-45x=0\\ \therefore & x(x-45)=0\end{array}$
By using the property, if the product of two numbers is zero, then at least one of them is zero, we get
$\begin{array}{cc}& \therefore x=0\text{or}x-45=0\\ & x=0\text{or}x=45\end{array}$
Length of rectangular farm
$$=2x+10=2\left(45\right)+10=100m$$
Side of the pond $=\frac{x}{3}=\frac{45}{3}=15m$
$\therefore$ Length and breadth of the farm and the side of the pond are $100m,45m$ and $15m$ respectively.