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Grade Upto college level Algebra

Given the sum of the perimeters of a square and a circle show that the sum of their areas is least when the side of the square is equal to the diameter of a circle.

Manvendra Singh chahar

12 Years agoGrade Upto college level

1 Answer

Jitender Singh

12 Years ago

Ans:

Let side of square to be ‘a’ & radius of circle to be ‘r’

Given:

$P = 4a + 2\pi r$

Sum of their area:

$A = a^{2}+\pi r^{2}$

$r = \frac{P-4a}{2\pi }$

$A = a^{2}+\pi (\frac{P-4a}{2\pi })^{2}$

$\frac{\partial A}{\partial a} = 2a+2\pi (\frac{P-4a}{2\pi })(\frac{-4}{2\pi }) = 0$

$\pi a = P - 4a$

$P = \pi a + 4a$

$\pi a+4a = 4a+2\pi r$

$a = 2r$

Thanks & Regards

Jitender Singh

IIT Delhi

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