Question

the rate of change of diagonal length r of a square with respect to its area A is ?​

Answer 1

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Answer 2

Given: A square.

To find: The rate of change of diagonal length r of a square with respect to its area A.

Solution:

• Let x be the length of the side of the square.

• The rate of change of the diagonal length with respect to the length of the side of the square can be written as,

        $\frac{dr}{dx} = \sqrt{2} x$

• This is so because the diagonal length is √2 times the side length in a square.
• The rate of change of the area with respect to the length of the side of the square can be written as,

        $\frac{dA}{dx} = x^{2}$

• This is so because the area is the square of the side length in a square.
• Now, the rate of change of diagonal length with respect to the area is calculated by dividing the two equations obtained.

        $\frac{dr/dx}{dA/dx} = \frac{\sqrt{2}x} {x^{2} }$

        $\frac{dr}{dA} = \frac{\sqrt{2}} {x}$

Therefore, the rate of change of diagonal length r of a square with respect to its area A is √2/x.