An architect designs a garden in a circular shape with centre E and radius 50 m. There is a flower bed ABCD of rectangular shape inside the garden as shown in the figure. Suppose length and width of the flower bed are 2x and 2y meters respectively.

Based on above information answer the following question:

The area of the flower bed (A(x)) is given by:

$4x\sqrt{2500-x^2}$

The correct answer is Option 2: $4x\sqrt{2500-x^2}$

• The garden is a circle with center E and radius = 50 m.

• The flower bed inside is a rectangle ABCD, centered at E.

• Dimensions of the rectangle:

  • Length = 2x,

  • Width = 2y

• Since the rectangle is centered at the origin (point E), the point A (–x, –y), B (–x, y), C (x, y), D (x, –y) all lie on the circle.

Step 1: Use the equation of the circle

• From the diagram, the diagonal of the rectangle (e.g., AC) is a chord of the circle that passes through the center — i.e., a diameter. Using Pythagoras Theorem in triangle AEC:

• x^2 + y^2 = 50^2 = 2500
• Thus, Y =$\sqrt{2500-x^2}$

Step 2: Area of rectangle (flower bed)

Area= L x B = (2x) (2y) = 4xy

Substitute the expression for y :

A (x) =4x $\sqrt{2500-x^2}$