Grade 12Algebra

1. A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then:

Narendra

8 Years agoGrade 12

1 Answer

Samyak Jain

8 Years ago

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units.

Perimeter of square = 4x & perimeter of circle (i.e. its circumference) = 2 $\pi$ r

So sum of perimeters of given figures is equal to 2,

i.e., 4x + 2 $\pi$ r = 2 or x = (1 – $\pi$ r)/2

Sum of areas of the square & the circle = x²+ $\pi$ r² , Put x = (1 – $\pi$ r)/2

= [(1 – $\pi$ r)/2]²+ $\pi$ r²

= (1 – 2 $\pi$ r + $\pi$ ² r²)/4 + $\pi$ r²

= (1/4)[1 – 2 $\pi$ r + $\pi$ ² r²+4 $\pi$ r²]

Differentiate the above expression wrt x and equate it to zero.

$\therefore$ (1/4)[0 – 2 $\pi$ + $\pi$ ² 2r + 4 $\pi$ .2r] = 0 => – 2 $\pi$ + $\pi$ ² 2r + 4 $\pi$ .2r = 0

2( $\pi$ ²+ 4 $\pi$ ) r = 2 $\pi$ => r = $\pi$ / ( $\pi$ ²+ 4 $\pi$ ) = 1/( $\pi$ + 4)

Differentiate (1/4)[0 – 2 $\pi$ + $\pi$ ² 2r + 4 $\pi$ .2r] to check the value of r we got is minimum or maximum.

We get (1/4)[0 – 0 + 2 $\pi$ ² + 8 $\pi$ ] > 0. Thus, minima is formed.

$\because$ x = (1 – $\pi$ r)/2 , x = (1 – $\pi$ / $\pi$ + 4)/2 = ( $\pi$ + 4 – $\pi$ ) / 2( $\pi$ + 4)

x = 4/2( $\pi$ + 4) = 2 / $\pi$ + 4

$\therefore$ x = 2 / ( $\pi$ + 4) & r = 1/( $\pi$ + 4) .

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1. A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then:

1. A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then:

Narendra

1 Answer

Samyak Jain

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units.

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