Geometry mathematics

Question

What is the largest length and breadth of a rectangle that can be kept inside a circle of diameter 80 with steps for calculation

Answer 1

First, I assume you meant what length and width produce the maximum area. Second, I assume you meant an inscribed rectangle; that is, one whose 4 vertices just touch the circle. Here's the procedure:

1. Draw a rectangle inscribed within a circle. The four vertices should just touch the circle
2. Draw a diameter across the circle through the middle of your rectangle
3. Draw a radius from the center of the circle to the upper right vertex of the rectangle. Label the angle between this radius and the diameter you drew as angle x.
4. The length and width of the rectangle can be expressed in terms of x as follows:

Length = 80 sin(x)

Width = 80 cos(x)

The area of the rectangle is:

Area = Length x Width = 80² sin(x) cos(x)

To find the maximum area, take the first derivative of Area wrt x, set it to zero, and solve for x. Once you have the value for x, plug it into the equations for Length and Width above.

Answer 2

Strictly speaking this question cannot be answered.

As length of the rectangle increases, width decreases.

The maximum occurs when length and width are both equal to 40√2.