## How should the wire of 28 m be cut so that the combined area of the circle and square is as small as possible ?

##
**Application of Derivative **

**A piece of wire 28 cm long is to be cut into two pieces. One piece is to be made into a circle and another into a square. How should the wire be cut so that the combined area of the two figures is as small as possible?**

Let the wire be cut at a distance of x meter from one end. Therefore then two pieces of wire be x m and (28-x) m.

## Calculate Dimension of Circle and Square

Since 1st part of the wire is turned into square. then its perimeter will be x m.

So using formula of perimeter of square , we can calculate side of the square = x/4 m

So using formula of perimeter of square , we can calculate side of the square = x/4 m

## Calculate Areas of Circle and Square

Therefore Area of square = (x/4)(x/4) sq m

A

_{1 }= x^{2}/16
And when 2nd part of the wire is turned to circle, then its perimeter ( circumference ) will be 28 - x m. So using formula of perimeter of square , And if "r" be radius of the circle , Then

Circumference of circle = 2 Ï€ r = (28-x)

∴ r = (28-x)/2Ï€

We know that Area of Circle A

_{2 }= Ï€ r^{2 }
A

##

_{2 }= Ï€[(28-x)/2Ï€]^{2 }^{}##
^{Express Areas in terms of Function}

^{}

^{}

##
^{To find value/s of x}

Now to find the value of x for which this function A(x) is maximum or minimum ,put A(x) = 0

^{}

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^{To Test the Minimum Value of Function}

^{}Now we have the value of "x" on which either A(x) have maximum or minimum value . To check the maximum or minimum value we have to find A''(x) as follows

^{}

^{}

^{}

So A''(x) has positive value Therefore A(x) shall have maximum value at x = 112/(Ï€ + 4)

Hence two pieces of wire should be of length x m and (28-x) m

These pieces should be of length 112/(Ï€+4) and 28Ï€/(Ï€ + 4)

##

^{Verification}

we can calculate the sum of these pieces , it must be 28 m

####
^{1st part }

^{ }

112/(Ï€+4) = 112/{(22/7)+4}=112×7/50 = 784/50

#### 2nd part

28Ï€/(Ï€ + 4) = {28×22/7}/{(22/7)+4} = 88×7/50 = 616/50

#### Sum of Two Parts

112×7/50 + 28×7/50 = (784+616)/50

= 1400/50= 28 m

^{}

^{}

## My previous Post

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**Quiz of Mathematics For You **

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**Do not Forget to watch this video of same Problem**

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You can clear your doubts if any after watching this video

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