Question

A plot is in the shape of a rectangle. The area of this rectangular plot is $200{{m}^{2}}$. If the length of this rectangular plot is 10m more than the breadth of the rectangular plot, find the length and breadth of the rectangular plot.

Answer 1

Hint: The area of a rectangle is equal to its length times its breadth. Since, it is given that length is 10 m more than breath, we can form an equation as l = 10 + b. Now, substituting this value of l in the formula of the area will give us a quadratic equation in b. We can then solve it by the method of splitting the middle term and get the value of b and hence the value of l.

Complete step by step answer:
We are given,
Area of the rectangular plot =$200{{m}^{2}}$
We are also given that the length of this rectangular plot is 10m more than the breadth of the rectangular plot. Let $l$ be the length of the plot and let $b$ be the breadth of the plot.
Then, mathematically speaking, we have
$l=10+b.....................(1)$

We know that the area of the rectangular plot = length $\times $ breadth. So, we can write it as
$=l\times b$
$\because l=10+b$
Area of the rectangular plot $=(10+b)\times b$
But, we know that the area of the rectangular plot = $200{{m}^{2}}$
$\therefore (10+b)\times b=200$
${{b}^{2}}+10b=200$
${{b}^{2}}+10b-200=0$
We have now obtained a quadratic equation.
We have three different methods to solve a quadratic equation namely factorizing or splitting the middle term, completing the square method and the quadratic formula (Using discriminant). Here we are solving the quadratic equation using factorizing method.
Using splitting the middle term method, as the name suggests we will split the middle term into two, whose sum is equal to the middle term and their product is equal to the product of the first and third term.
In the quadratic equation ${{b}^{2}}+10b-200=0$, the product is equal to $-200{{b}^{2}}$ and the sum is equal to $10b$.
Hence, we split the equation as,
${{b}^{2}}+20b-10b-200=0$
$b(b+20)-10(b+20)=0$
$(b-10)(b+20)=0$
$\therefore b-10=0$ and $b+20=0$
Hence, the values of b are 10 and -20.
Since, measurement of lengths cannot take a negative value, we reject the value of $b=-20$.
$\therefore $ Breadth of the rectangular plot, $b=10m$
Substituting $b=10m$ in equation (1), we get,
 $l=10+b=10+10=20m$
$\therefore $ Length of the rectangular plot, $l=20m$
Hence, the length and breadth of the rectangular plot are 20m and 10m respectively.
Note:
Alternate method:
We could have substituted, $b=l-10$ in,
Area of the rectangular plot $=l\times b$
On substituting for $b$ and area, the above equation becomes,
$l\times (l-10)=200$
${{l}^{2}}-10l-200=0$
Solving the above quadratic equation, we obtain the value of $l$ and hence $b$.