Question

A square is inscribed in a circle. How fast is the area of the square changing when the area of the circle is increasing at the rate of 1 square inch per minute?

Answer 1

The rate at which the area of the inscribed square is changing with respect to time is 1/2 square inches per minute.

Step-by-step explanation:

Let's denote the side length of the square as s, and the radius of the circle as r. In an inscribed square, the diagonal of the square is equal to the diameter of the circle, which is twice the radius. Therefore, s = √2 * r.

Now, we can express the area of the square (Aₛ) in terms of its side length:

Aₛ = s² = (√2 * r)² = 2 * r²

Differentiate both sides with respect to time (t):

dAₛ/dt = 4 * r * dr/dt

Given that dAₙ/dt = 1 square inch per minute (where Aₙ is the area of the circle), and the area of the circle (Aₙ) is given by Aₙ = π * r², we can find dr/dt:

dr/dt = 1 / (2πr)

Now, substitute this into the expression for dAₛ/dt to find the rate of change of the square's area:

dAₛ/dt = 4 * r * (1 / (2πr)) = 1 / (2π)

Therefore, the rate at which the area of the inscribed square is changing is 1/(2π) square inches per minute.