10. A piece of wire 28 cm in length is cut into two parts, each part bend to make a square. Given that area of
square is 416 m .find the perimeter of each square.
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Step-by-step explanation:
Let the 2 pieces of rope be x, (28−x)m
Area of square =(28−x)
2
Area of triangle =
4
3
x
2
Total Area =(28−x)
2
+
4
3
x
2
Differentiating wrt x:-
dx
dy
=2(28−x)(−1)+
2
3
x
dx
2
d
2
y
=−2(−1)+
2
3
=−3+
2
3
=−ve
Since
dx
2
d
2
y
is negative, therefore maxima will be obtained.
To find critical points,
dx
dy
=0
⇒2(28−x)=
2
3
x
⇒4×28=(
3
+2)x
⇒x=
3
+2
112
[value of x where maxima will be obtained
rope used in square
=28−x=28−
3
+2
112
=
3
+2
28
3
−56
=
(
3
+2)(
3
−2)
28(
3
−2)(
3
−2)
=
−1
28[3+4−4
3
]
=28(4
3
−7)m.
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