The area of a rectangle and the square of its perimeter are in the ratio 1 ∶ 25. Then the lengths of the shorter and longer sides of the rectangle are in the ratio
Let 'a' and 'b' be the length of sides of the rectangle. (a > b)
Area of the rectangle = a*b
Perimeter of the rectangle = 2*(a+b)
$$\Rightarrow$$ $$\dfrac{a*b}{(2*(a+b))^2}=\dfrac{1}{25}$$
$$\Rightarrow$$ $$25ab=4(a+b)^2$$
$$\Rightarrow$$ $$4a^2-17ab+4b^2=0$$
$$\Rightarrow$$ $$(4a-b)(a-4b)=0$$
$$\Rightarrow$$ $$a = 4b$$ or $$\dfrac{b}{4}$$
We initially assumed that a > b, therefore a $$\neq$$ $$\dfrac{b}{4}$$.
Hence, a = 4b
$$\Rightarrow$$ b : a = 1 : 4
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