Certainly! Let's solve this problem step by step:
1. Expression for Area:
The given area of the rectangular field is [tex]\( x^2 - 8x + 15 \)[/tex] square meters.
2. Factor the Area Expression:
To find the dimensions (length and breadth), we need to factor the quadratic expression [tex]\( x^2 - 8x + 15 \)[/tex]. Factoring this expression, we get:
[tex]\[
x^2 - 8x + 15 = (x - 5)(x - 3)
\][/tex]
3. Identify the Length and Breadth:
Since the factored form represents the product of two binomials, each corresponding to a dimension of the rectangle, we identify:
[tex]\[
\text{Length} = x - 5
\][/tex]
[tex]\[
\text{Breadth} = x - 3
\][/tex]
4. Calculate the Perimeter:
The perimeter [tex]\( P \)[/tex] of a rectangle is given by:
[tex]\[
P = 2(\text{Length} + \text{Breadth})
\][/tex]
Substituting the length and breadth we found:
[tex]\[
P = 2((x - 5) + (x - 3))
\][/tex]
[tex]\[
P = 2(x - 5 + x - 3)
\][/tex]
Combine like terms inside the parentheses:
[tex]\[
P = 2(2x - 8)
\][/tex]
Simplify:
[tex]\[
P = 4x - 16
\][/tex]
5. Summary:
- The length of the field is [tex]\( x - 5 \)[/tex].
- The breadth of the field is [tex]\( x - 3 \)[/tex].
- The perimeter of the field is [tex]\( 4x - 16 \)[/tex].
So, the dimensions of the rectangular field and its perimeter are:
- Length: [tex]\( x - 5 \)[/tex]
- Breadth: [tex]\( x - 3 \)[/tex]
- Perimeter: [tex]\( 4x - 16 \)[/tex] meters
