Solving for Rectangular Pitch Breadth
This solution explains how to find the breadth of a rectangular pitch given its area and the relationship between its length and breadth.
Understanding the Problem
We are given a rectangular pitch with the following information:
- The length ($l$) is 30 m more than its breadth ($b$). This can be written as: $l = b + 30$
- The area ($A$) of the pitch is $18,271$ m$^2$.
- We need to find the breadth ($b$) of the pitch in meters.
Setting Up the Equation
The formula for the area of a rectangle is:
$A = \text{length} \times \text{breadth}$
Substituting the given values and the relationship between length and breadth:
$18,271 = (b + 30) \times b$
Solving the Quadratic Equation
Expanding the equation, we get:
$18,271 = b^2 + 30b$
Rearranging this into the standard quadratic equation form ($ax^2 + bx + c = 0$):
$b^2 + 30b - 18,271 = 0$
We can solve this quadratic equation for '$b$' using the quadratic formula:
$b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$
In our equation, $A=1$, $B=30$, and $C=-18,271$. Plugging these values into the formula:
$b = \frac{-30 \pm \sqrt{30^2 - 4(1)(-18,271)}}{2(1)}$
$b = \frac{-30 \pm \sqrt{900 + 73,084}}{2}$
$b = \frac{-30 \pm \sqrt{73,984}}{2}$
To find the value of $\sqrt{73,984}$: $\sqrt{73,984} = 272$
Now, substitute this back into the equation for '$b$':
$b = \frac{-30 \pm 272}{2}$
Determining the Breadth
We have two possible solutions for '$b$':
- $b = \frac{-30 + 272}{2} = \frac{242}{2} = 121$
- $b = \frac{-30 - 272}{2} = \frac{-302}{2} = -151$
Since the breadth of a rectangle must be a positive value, we discard the negative solution.
Therefore, the breadth ($b$) of the rectangular pitch is 121 m.
Verification
If the breadth is 121 m, then the length is:
$l = b + 30 = 121 + 30 = 151$ m
The area calculated using these dimensions is:
$A = l \times b = 151 \times 121 = 18,271$ m$^2$
This matches the given area, confirming our calculation.
Final Answer
The breadth of the rectangular pitch is 121 m.
