Question

A rectangular plot of length 8 m excess of its breadth. Its area is $308{m^2}$ . Find length, breadth and its perimeter.

Answer 1

Hint: First calculate the length and breadth of plot by assuming any one of the dimensions and using the given conditions. Then calculate the perimeter of the rectangular plot.

Complete step-by-step answer:

Given the problem, a rectangular plot of length 8 m excess of its breadth.
Also, its area is $308{m^2}$.

Let the length of the rectangular plot be $x$ meters.

Since the length of the rectangular plot exceeds its breadth by 8 metres.

From the given condition, breadth of the rectangular plot will be $\left( {x - 8} \right)$ metres.

It is given that the plot is in the form of a rectangle and we know that area of rectangle is given by,
Area $ = {\text{length}} \times {\text{breadth}}$

Using the assumed dimensions of the rectangular plot, we get
Area $ = x \times \left( {x - 8} \right) = {x^2} - 8x$

Also, since the area of the rectangular plot is given to be $308{m^2}$.
$
   \Rightarrow 308 = {x^2} - 8x \\
   \Rightarrow {x^2} - 8x - 308 = 0 \\ $........................(1)

Above equation is of the type
$a{x^2} + bx + c = 0$.................................(2)
Where, $\left(
  a = 1, b = - 8, c = - 308 \right)$.........................(3)
Roots of equation $(2)$ are given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Using $(3)$in above formula, we get
\[
  x = \dfrac{{8 \pm \sqrt {64 + 1232} }}{2} = \dfrac{{8 \pm \sqrt {1296} }}{2} = \dfrac{{8 \pm 36}}{2} \\
   \Rightarrow x = 22, - 14 \\
\]

Since $x$ is the length of the plot which cannot be a negative quantity,
\[ \Rightarrow x = 22\] metre is the required solution of equation $(1)$.

Hence length of the rectangular plot is equal to $x = 22$ metre.

Breadth of the rectangular plot is equal to $\left( {x - 8} \right) = 14$ metre.

Also, we know that the perimeter of a rectangle is given by $2({\text{Length + Breadth)}}$ metres.
Using the obtained values of length and breadth in above, we get
Perimeter$ = 2\left( {22 + 14} \right) = 2 \times 36 = 72$ metres.
Hence the perimeter of the rectangular plot is $72$ metres.

Note: The formulae for area and perimeter of the rectangle should be kept in mind for solving problems like above. Also, the method to solve quadratic equations using the discriminant method is advised for better accuracy.