The ratio of the length and perimeter of a rectangle is 7:22, respectively. If the area of the rectangle is 1,792 cm², then find the breadth of the rectangle.
32 cm
Let the length be \(l\) and the breadth be \(b\). The perimeter of a rectangle is \(2(l + b)\).
Given length : perimeter \(= 7 : 22\), so \(\dfrac{l}{2(l+b)} = \dfrac{7}{22}\).
Cross-multiply: \(22l = 14(l + b) \Rightarrow 22l - 14l = 14b \Rightarrow 8l = 14b \Rightarrow l = \dfrac{7b}{4}\).
The area is \(l \times b = \dfrac{7b}{4}\times b = \dfrac{7b^2}{4}\), and this equals \(1792\ \text{cm}^2\).
Solve for \(b\): \(b^2 = \dfrac{1792 \times 4}{7} = \dfrac{7168}{7} = 1024 \Rightarrow b = \sqrt{1024} = 32\ \text{cm}\).
Key concept: converting the given ratio into a relation between \(l\) and \(b\) reduces the area equation to a single variable.
Hence the breadth of the rectangle is 32 cm.
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