Question

Following diagram shows the atmosphere of two different places. Analyze the environments of both places Fig 1: _____________________________________________ Fig 2: _____________________________________________​

Answer 1

Answer★

Given :-

The perimeter of an aluminium sheet is 120 cm, if its length is reduced by 10% and its breadth is increased by 20% the perimeter does not change.

To Find:-

What is the measure of the length and the breadth of the sheet.

Formula Used :-

$\clubsuit$ Perimeter Of Rectangle Formula :

$\footnotesize\mapsto \sf\boxed{\bold{\pink{Perimeter_{(Rectangle)} =\: 2(Length + Breadth)}}}$

Solution:-

Let,

$\mapsto \bf Length_{(Sheet)} =\: x\: cm$

$\mapsto \bf Breadth_{(Sheet)} =\: y\: cm$

$\clubsuit\: \: \sf\bold{\purple{\underline{In\: the\: 1^{st}\: case\: :-}}}$

Given :

Perimeter of an aluminium sheet = 120 cm

According to the question by using the formula we get,

$\implies \sf 120 =\: 2(x + y)$

$\implies \sf \dfrac{\cancel{120}}{\cancel{2}} =\: x + y$

$\implies \sf 60 =\: x + y$

$\implies \sf\bold{\green{x + y =\: 60\: ------\: (Equation\: No\: 1)}}$

$\clubsuit\: \: \sf\bold{\purple{\underline{In\: the\: 2^{nd}\: case\: :-}}}$

$\bigstar$ The length is reduced by 10% and its breadth is increased by 20%, the perimeter does not change.

According to the question,

$\implies \sf 2(x - 10\% \times x + y + 20\% \times y) =\: 2(x + y)$

$\implies \sf 2\bigg(x - \dfrac{1\cancel{0}}{10\cancel{0}} \times x + y + \dfrac{2\cancel{0}}{10\cancel{0}} \times y\bigg) =\: 2\bigg(x + y\bigg)$

$\implies \sf 2\bigg(x - x \times \dfrac{1}{10} + y + y \times \dfrac{\cancel{2}}{\cancel{10}}\bigg) =\: 2\bigg(x + y\bigg)$

$\implies \sf 2\bigg(x - \dfrac{x}{10} + y + y \times \dfrac{1}{5}\bigg) =\: 2\bigg(x + y\bigg)$

$\implies \sf {\cancel{2}}\bigg(x - \dfrac{x}{10} + y + \dfrac{y}{5}\bigg) =\: {\cancel{2}}\bigg(x + y\bigg)$

$\implies \sf \dfrac{9x}{10} + \dfrac{7y}{5} =\: x + y$

$\implies \sf \dfrac{9x + 14y}{10} =\: x + y$

By doing cross multiplication we get,

$\implies \sf 9x + 14y =\: 10(x + y)$

$\implies \sf 9x + 14y =\: 10x + 10y$

$\implies \sf 10x - 9x + 10y - 14y =\: 0$

$\implies \sf\bold{\green{x - 4y =\: 0\: ------\: (Equation\: No\: 2)}}$

By solving the equation no 1 and 2 we get,

$\implies \sf x + y - (x - 4y) =\: 60 - 0$

$\implies \sf x + y - x + 4y =\: 60$

$\implies \sf {\cancel{x}} {\cancel{- x}} + y + 4y =\: 60$

$\implies \sf y + 4y =\: 60$

$\implies \sf 5y =\: 60$

$\implies \sf y =\: \dfrac{\cancel{60}}{\cancel{5}}$

$\implies \sf\bold{\blue{y =\: 12}}$

By putting the value of y in the equation no 2 we get,

$\implies \sf x - 4y =\: 0$

$\implies \sf x - 4(12) =\: 0$

$\implies \sf x - 48 =\: 0$

$\implies \sf\bold{\blue{x =\: 48}}$

Hence, the required length and breadth of the sheet are :

$\longrightarrow \sf\bold{\red{Length_{(Sheet)} =\: 48\: cm}}$

$\longrightarrow \sf\bold{\red{Breadth_{(Sheet)} =\: 12\: cm}}$

${\small{\bold{\underline{\therefore\: The\: length\: and\: breadth\: of\: sheet\: is\: 48\: cm\: and\: 12\: cm\: respectively\: .}}}}$