# Rectangle - area, perimeter

The area of a rectangular field is equal to 300 square meters. Its perimeter is equal to 70 meters. Find the length and width of this rectangle.

Correct result:

a =  20 m
b =  15 m

#### Solution:

$S=300 \ \text{m}^2 \ \\ p=70 \ \text{m} \ \\ S=ab \ \\ p=2(a+b) \ \\ \ \\ 300=ab \ \\ 35=a+b \ \\ \ \\ a(35-a)=300 \ \\ \ \\ -a^2 +35a -300=0 \ \\ a^2 -35a +300=0 \ \\ \ \\ p=1; q=-35; r=300 \ \\ D=q^2 - 4pr=35^2 - 4\cdot 1 \cdot 300=25 \ \\ D>0 \ \\ \ \\ a_{1,2}=\dfrac{ -q \pm \sqrt{ D } }{ 2p }=\dfrac{ 35 \pm \sqrt{ 25 } }{ 2 } \ \\ a_{1,2}=\dfrac{ 35 \pm 5 }{ 2 } \ \\ a_{1,2}=17.5 \pm 2.5 \ \\ a_{1}=20 \ \\ a_{2}=15 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ (a -20) (a -15)=0 \ \\ \ \\ a=a_{1}=20 \ \text{m}$

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