A square is inscribed in a circle. How fast is the area between the square and the circle changing when the area of the circle is increasing at the rate of one square inch per minute?
I'm at loss as to how to solve this problem. Any help?
A square is inscribed in a circle. How fast is the area between the square and the circle changing when the area of the circle is increasing at the rate of one square inch per minute?
I'm at loss as to how to solve this problem. Any help?
A good start is to draw a picture. (And assume that the square is expanding along with the circle!)
Draw the square's diagonals; these are also the diameters of the circle. Label a half-diameter as "r", the radius.
What is the area of a circle with radius r?
By its nature, the diameters of the circle split the square into four isosceles triangles with base-angles of 45 degrees. What is the length of one of the equal-length sides of one of these right triangles? What then is the length, in terms of r, of a side of the square, being the hypotenuse of the triangle?
What is the area of a square with this side length?
Since the square is inside the circle, subtract the square's area from the circle's area to find an expression for what you need.
Then differentiate with respect to time, etc, etc.
If you get stuck, please reply showing how far you have gotten. Thank you!
Just before I'm supposed to differentiate, I have A = pi(r)^2 - 2(r)^2, which expresses the area between the circle and the square. I could've easily differentiated the expression but the problem is that I have no idea what dr/dt and "r" are supposed to be.