Question

A rectangle is inscribed in a circle of radius 5 inches. if the length of the rectangle is decreasing at the rate of 2 inches per second, how fast is the area of the rectangle changing at the instant when the length is 6 inches?

Answer 1

The rate at which the area of the rectangle is changing can be found using the relationship between the length, width, and area of the rectangle. By differentiating both sides of the equation representing the rectangle's properties, we can solve for the rate of change of the width when the length is 6 inches.

Step-by-step explanation:

To find the rate at which the area of the rectangle is changing, we can use the relationship between the length, width, and area of the rectangle. Let's denote the length of the rectangle as 'L' and the width as 'W'. Since the rectangle is inscribed in a circle, the diagonal of the rectangle is equal to the diameter of the circle, which is 2 times the radius. So we have the equation: L^2 + W^2 = (2 * 5)^2 = 100.

Now, we can differentiate both sides of the equation with respect to time t: 2L * dL/dt + 2W * dW/dt = 0, since the total area A = L * W is constant.

Substituting the value of L = 6 inches and dL/dt = -2 inches/second, we can solve for dW/dt to find the rate at which the width is changing when the length is 6 inches.