I’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 $P(x)=4x$. And the rate of change of the area wrt its side is $\frac{dA}{dx}=2x$. With a circle, the area as a function of the radius is $A(r)=\pi(r^2)$. And the rate of change of the area wrt its radius is $\frac{dA}{dr}=2\pi(r)$. The circumference as a function of the radius is also $C(r)=2\pi(r)$. 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, $\frac{dA}{dx}=2x=\frac{4x}{2}$.
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, $A(r)=\pi(\frac{x}{2})^2=\frac{\pi}{4}x^2$ and $C(\frac{x}{2})=2\pi(\frac{x}{2}$, and that $\frac{dA}{dr}=\frac{\pi}{2}x$.
Somehow 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.
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.