Question

The perimeter of a rectangle and a square are $160m$ each. The area of the rectangle is less than that of the square by $100$ sq.m. The length of the rectangle is:

• A. $30m$
• B. $60m$
• C. $40m$
• D. $50m$

Answer 1

The correct option is D $50m$

Perimeter of square $=4a$

$\Rightarrow 160=4a$

$\Rightarrow a=\frac{160}{4}$

$\Rightarrow a=40m$

Perimeter of rectangle $=2(l+b)$

$\Rightarrow 160=2(l+b)$

$\Rightarrow l+b=\mathrm{80.....}\left(i\right)$

As per question

Area of square $-$ Area of rectangle $=100$

$a^2-lb=100$

${40}^2-lb=100$

$lb=1600-100$

$lb=\mathrm{1500......}\left(ii\right)$

From (i) and (ii)

$\Rightarrow \frac{1500}{b}+b=80$

$\Rightarrow 1500+b^2=80b$

$\Rightarrow b^2-80b+1500=0$

$\Rightarrow b^2-50b-30b+1500=0$

$\Rightarrow b(b-50)-30(b-50)=0$

$\Rightarrow (b-50)(b-30)=0$

$\Rightarrow b=50,30$

If $b=50m$ then $l+50=80\Rightarrow l=30m$

If $b=30m$ then $l+30=80\Rightarrow l=50m$

As length of the rectangle is more than breadth so $l=50m$ and $b=30m$