A rectangular carpet has an area of 168 $$\text{m}^2$$​ and a perimeter of 62 m. The length of its diagonal is:

Correct option is B

Given:

Area of the rectangle = 168 m²

Perimeter of the rectangle = 62 m

Formula Used:

Area = length × breadth

Perimeter = 2(length + breadth)

Diagonal = $$\sqrt{(\text{length² + breadth²})}$$​​

Solution:
Let length = l and breadth = b

From perimeter:

2(l + b) = 62

​$$l + b = 31 \quad \text{(1)}$$​

From area:

​$$l \cdot b = 168 \quad \text{(2)}$$​

From equations (1) and (2), solve for l and b using substitution:

Let b = $$31 - l$$​​
Substitute in (2):

​$$l(31 - l) = 168$$​

​$$31l - l^2 = 168$$​

​$$l^2 - 31l + 168 = 0$$​

l $$= \frac{31 \pm \sqrt{31^2 - 4 \cdot 168}}{2} = \frac{31 \pm \sqrt{961 - 672}}{2} = \frac{31 \pm \sqrt{289}}{2} = \frac{31 \pm 17}{2}$$​

So,

$$l = \frac{48}{2} = 24 \quad \text{or} \quad l = \frac{14}{2} = 7$$

Therefore, the dimensions are:
​$$l = 24 \, \text{m}, \quad b = 7 \, \text{m}$$​​
(or vice versa, since length and width are interchangeable).
​Diagonal$$= \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \, \text{m}$$ 

Alternate Method:​

​$$l + b = 31 \quad \text{(1)}$$​​

​$$l \cdot b = 168 \quad \text{(2)}$$

l² + b²= ( l + b)² - 2lb

l² + b²= 31² - 2$$\times$$ 168

​l² + b²= 961 - 336 = 625

Now,

​$$​\sqrt{l^2 + b^2}=\sqrt{625}$$ = 25 m​

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