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Question Number 33805 by pieroo last updated on 25/Apr/18

The perimeter of a square and a rectangle is the same. The width of the rectangle is 6 cm and its area is 16 cm^2 less than the area of the square. Find the area of the square.

$$\mathrm{The}\:\mathrm{perimeter}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{and}\:\mathrm{a}\:\mathrm{rectangle}\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{same}.\:\mathrm{The}\:\mathrm{width}\:\mathrm{of}\:\mathrm{the}\:\mathrm{rectangle}\:\mathrm{is}\:\mathrm{6}\:\mathrm{cm}\:\mathrm{and}\:\mathrm{its}\:\mathrm{area} \\ $$$$\mathrm{is}\:\mathrm{16}\:\mathrm{cm}^{\mathrm{2}} \:\mathrm{less}\:\mathrm{than}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}. \\ $$

Answered by MJS last updated on 25/Apr/18

square=a×a rectangle=6×b 6b+16=a^2 4a=2b+12 ⇒ b=2a−6 6(2a−6)+16=a^2 12a−20=a^2 a^2 −12a+20=0 (a−2)(a−10)=0 a_1 =2 ⇒ b_1 =2a_1 −6=−2 not valid a_2 =10 ⇒ b_2 =2a_2 −6=14 area of the square=a_2 ^2 =100cm^2

$$\mathrm{square}={a}×{a} \\ $$$$\mathrm{rectangle}=\mathrm{6}×{b} \\ $$$$ \\ $$$$\mathrm{6}{b}+\mathrm{16}={a}^{\mathrm{2}} \\ $$$$\mathrm{4}{a}=\mathrm{2}{b}+\mathrm{12}\:\Rightarrow\:{b}=\mathrm{2}{a}−\mathrm{6} \\ $$$$ \\ $$$$\mathrm{6}\left(\mathrm{2}{a}−\mathrm{6}\right)+\mathrm{16}={a}^{\mathrm{2}} \\ $$$$\mathrm{12}{a}−\mathrm{20}={a}^{\mathrm{2}} \\ $$$${a}^{\mathrm{2}} −\mathrm{12}{a}+\mathrm{20}=\mathrm{0} \\ $$$$\left({a}−\mathrm{2}\right)\left({a}−\mathrm{10}\right)=\mathrm{0} \\ $$$${a}_{\mathrm{1}} =\mathrm{2}\:\Rightarrow\:{b}_{\mathrm{1}} =\mathrm{2}{a}_{\mathrm{1}} −\mathrm{6}=−\mathrm{2}\:\mathrm{not}\:\mathrm{valid} \\ $$$${a}_{\mathrm{2}} =\mathrm{10}\:\Rightarrow\:{b}_{\mathrm{2}} =\mathrm{2}{a}_{\mathrm{2}} −\mathrm{6}=\mathrm{14} \\ $$$$\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}={a}_{\mathrm{2}} ^{\mathrm{2}} =\mathrm{100}{cm}^{\mathrm{2}} \\ $$

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