The area of the rectangle below is 96 cm2. find x
Answers
Step-by-step explanation:
Let length and bredth of rectangle are x and y units respectively. Area =x×y.
x×y=96 => y= 96/x
P=2(l+b)
p=2(x+y)
p=2(x+96/x)=2x+192(1/x)
dp/dx=2–192/x^2. For minimum perimeter put dp/dx=0.
0=2–192/x^2
or 192/x^2=2
or 2x^2=192
or x^2=96
or x= 4.(6)^1/2 units
y=96/x=96/4.(6)^1/2= 4.(6)^1/2 unit
d2p^dx^2=2+384/x^3= + ve
There exist minima at x=4.(6)^1/2units.
In case minimum perimeter , each side of rectangle is 4.6^1/2 units i,e a square.
Minimum perimeter=2(4.6^1/2+4.6^1/2) units.
= 16.(6)^1/2 units. ,
Other - Cases:-
If , Area , x , y , p=2(x+y)
96=12×8 , 12 , 8 ,p=2(12+8)=40 units.
96=16×6 , 16 , 6 ,p=2(16+6)=44 units.
96=24×4 , 24 , 4 ,p=2(24+4)=56units
96=32×3 , 32 , 3 ,p=2(32+3)=70 units.
96=48×2 , 48 ,2 , p=2(48+2)=100 units.
96=96×1 , 96 , 1 , p=2(96+1)=194 units.
96=960×.1, 960 , .1, p=2(960+.1)=1920.2units
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