A circle is inside a square.

Question

A circle is inside a square.

The radius of the circle is increasing at a rate of 4 meters per hour and the sides of the square are increasing at a rate of 1 meter per hour.

When the radius is 2 meters, and the sides are 21 meters, then how fast is the AREA outside the circle but inside the square changing?

The rate of change of the area enclosed between the circle and the square is ___  square meters per hour.

Answer 1

This sounds like a related rates problem, so let's come up with a formula for the area of enclosed between the square and circle:

A = (area of square) - (area of circle)

A = s² - πr²

^{This means the formula for the rate at which the area changes is:}

^{A' = 2s * s' - 2πr * r'}

^{Now, we plug in the following information:}

^{s = 21m}

^{s' = 1 m/h}

^{r = 2m}

^{r' = 4 m/h}

^{This gives us:}

^{A' = 2(21)(1) - 2(π)(2)(2)}

^{A' ≈ 42 - 18.85}

A' ≈ 23.15 m²/h

The rate of change of the area enclosed between the circle and the square is 23.15 square meters per hour.