A rectangle 4

A rectangle has area 300 and perimeter 80. what is the ratio of the length and width?

Correct answer:

r =  3:1

Step-by-step explanation:

$$\begin{gathered} A=300 \\ p=80 \\ A=ab \\ p=2(a+b) \\ b=p\mathrm{/}2-a \\ A=a\ast (p\mathrm{/}2-a) \\ 300=a\cdot (80\mathrm{/}2-a) \\ {a}^2-40a+300=0 \\ p=1;q=-40;r=300 \\ D=q^2-4pr=40^2-4\cdot 1\cdot 300=400 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{40\pm \sqrt{400}}{2}} \\ {a}_{1,2}={\displaystyle \frac{40\pm 20}{2}} \\ {a}_{1,2}=20\pm 10 \\ {a}_1=30 \\ {a}_2=10 \\ \text{ Factored form of the equation: } \\ (a-30)(a-10)=0 \\ a=a_1=30 \\ b=p\mathrm{/}2-a=80\mathrm{/}2-30=10 \\ r=a\mathrm{/}b=30\mathrm{/}10=3=3:1 \end{gathered}$$

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