A rectangle of area 105 cm² has its length equal to x cm. Write down its breadth in terms of x. Given that the perimeter is 44 cm, write down an equation in x and solve it to determine the dimensions of rectangle.

Length of rectangle = x

Since the area of rectangle = 105 cm², breadth = $\dfrac{105}{x}$ cm.

∴ Perimeter = 2(length + breadth)

$= 2(x + \dfrac{105}{x}) \\[0.5em] = 2(\dfrac{x^2 + 105}{x}) \\[0.5em]$

Given, Perimeter = 44 cm

$\Rightarrow 2(\dfrac{x^2 + 105}{x}) = 44 \\[0.5em] \Rightarrow \dfrac{x^2 + 105}{x} = 22 \\[0.5em] \Rightarrow x^2 + 105 = 22x \\[0.5em] \Rightarrow x^2 - 22x + 105 = 0 \\[0.5em] \Rightarrow x^2 - 15x - 7x + 105 = 0 \\[0.5em] \Rightarrow x(x - 15) - 7(x - 15) = 0 \\[0.5em] \Rightarrow (x - 7)(x - 15) = 0 \\[0.5em] \Rightarrow x - 7 = 0 \text{ or } x - 15 = 0 \\[0.5em] \Rightarrow x = 7 \text{ or } x = 15$

Breadth = $\dfrac{105}{x}$ cm
Equation in x for perimeter, 2(x + $\dfrac{105}{x}$) = 44
Length = 7 cm , Breadth = 15 cm