Rates of changeI’m having some trouble with part c) of the following questions,a) What is the rate of change of the area A of a square with respect to its side x?b) What is the rate of change of the area A of a circle with respect to its radius r?c) Explain why one answer is the perimeter of the figure but the other answer is not.So, knowing that if we have a square with side length x, then the area of the square as a function of its side is A ( x ) = x 2 . The perimeter as a function of the side is P ( x ) = 4 x. And the rate of change of the area wrt its side is d A d x = 2 x. With a circle, the area as a function of the radius is A ( r ) = π ( r 2 ). And the rate of change of the area wrt its radius is d A d r = 2 π ( r ). The circumference as a function of the radius is also C ( r ) = 2 π ( r ). Therefore it’s the circle that’s the figure with the rate of change of the area wrt its radius equal to its perimeter, and what I saw was that the square had a rate of change of area wrt its side equal to half the perimeter of the square, d A d x = 2 x = 4 x 2 I inscribed a circle in a square with radius equal to half the square’s side length and went through the same work and then arrived at this, A ( r ) = π ( x 2 ) 2 = π 4 x 2 and C ( x 2 ) = 2 π ( x 2 and that d A d r = π 2 xSomehow in this example, I don’t think it’s correct because the same fact about the rate of change of area wrt radius being equal to perimeter doesn’t hold. I appreciate any help in explaining this, thank you.

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Rates of changeI’m having some trouble with part c) of the following questions,a) What is the rate of change of the area A of a square with respect to its side x?b) What is the rate of change of the area A of a circle with respect to its radius r?c) Explain why one answer is the perimeter of the figure but the other answer is not.So, knowing that if we have a square with side length x, then the area of the square as a function of its side is   A  (  x  )  =      x    2  . The perimeter as a function of the side is   P  (  x  )  =  4  x. And the rate of change of the area wrt its side is             d      A              d      x        =  2  x. With a circle, the area as a function of the radius is   A  (  r  )  =  π  (      r    2    ). And the rate of change of the area wrt its radius is             d      A              d      r        =  2  π  (  r  ). The circumference as a function of the radius is also   C  (  r  )  =  2  π  (  r  ). Therefore it’s the circle that’s the figure with the rate of change of the area wrt its radius equal to its perimeter, and what I saw was that the square had a rate of change of area wrt its side equal to half the perimeter of the square,             d      A              d      x        =  2  x  =            4      x        2  I inscribed a circle in a square with radius equal to half the square’s side length and went through the same work and then arrived at this,   A  (  r  )  =  π  (      x    2        )    2    =      π    4        x    2   and   C  (      x    2    )  =  2  π  (      x    2   and that             d      A              d      r        =      π    2    xSomehow in this example, I don’t think it’s correct because the same fact about the rate of change of area wrt radius being equal to perimeter doesn’t hold. I appreciate any help in explaining this, thank you.

Kyle George · Accepted Answer

The difference between the circle and the square is that when you increase the radius of a circle the increase goes all around the circle but when you increase the side of a square the increase is divided among two sides and only half of the increase goes around.So the dr multiplies by the circumference but only half of dx multiplies by the perimeter.

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