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

Given,

Area of rectangle = 105 cm²

Length of rectangle = x cm

By formula,

Area of rectangle = length × breadth

Substituting the values we get,

105 = x × breadth

Breadth = $\dfrac{105}{x}$ cm.

Perimeter of rectangle = 44 cm

$\therefore 2(l + b) = 44 \\[1em] \Rightarrow 2\Big(x + \dfrac{105}{x}\Big) = 44 \\[1em] \Rightarrow \Big(x + \dfrac{105}{x}\Big) = 22 \\[1em] \Rightarrow \dfrac{x^2 + 105}{x} = 22 \\[1em] \Rightarrow x^2 + 105 = 22x \\[1em] \Rightarrow x^2 - 22x + 105 = 0 \\[1em] \Rightarrow x^2 - 15x - 7x + 105 = 0 \\[1em] \Rightarrow x(x - 15) - 7(x - 15) = 0 \\[1em] \Rightarrow (x - 7)(x - 15) = 0 \\[1em] \Rightarrow x - 7 = 0 \text{ or } x - 15 = 0 \\[1em] \Rightarrow x = 7 \text{ or } x = 15.$

If x = 7 cm,

Breadth = $\dfrac{105}{7}$ = 15 cm

If x = 15 cm,

Breadth = $\dfrac{105}{15}$ = 7 cm

Hence, breadth = $\dfrac{105}{x}$, equation : 44 = $2\Big(x + \dfrac{105}{x}\Big)$ and the required dimensions of rectangle are 15 cm and 7 cm.