a rectangle PQRS inscribed in a quadrant of circle of radius 13cm. find area of rectangle.

Place the circle with centre at (0,0).

So rectangle has vertices at (0,0), (x,0), (x,y) and (0,y)
area = xy
but x^2 +y^2 = 13^2
y = √(169 - x^2)

area = x√(169-x^2)

If you want the MAXIMUM area, you would now differentiate that , set it equal to zero, and solve for x

To find the area of the rectangle PQRS inscribed in a quadrant of a circle, we need to first determine the dimensions of the rectangle.

Let's assume that the point Q is the midpoint of the arc RS. Hence, the length of the arc RS is half the circumference of the circle.

Circumference of a circle = 2πr (where r is the radius)

So, the length of the arc RS = (1/2) × 2πr = πr

Since the radius of the circle is given as 13 cm, the length of the arc RS is π × 13 cm.

Now, the length of the arc RS is equal to the length of the rectangle PQ. Hence, the length of the rectangle PQ = π × 13 cm.

Since the rectangle PQRS is inscribed in the quadrant of the circle, the length of the rectangle PS will be equal to the radius of the circle, which is 13 cm.

Therefore, we have:
Length of PQ = π × 13 cm
Length of PS = 13 cm

To calculate the area of the rectangle, we multiply the length and width:
Area of rectangle PQRS = Length × Width

Area of rectangle PQRS = (π × 13 cm) × (13 cm)

Using a calculator, we can approximate the value as:
Area of rectangle PQRS ≈ 530.929158 cm² (rounded to three decimal places)

Therefore, the approximate area of the rectangle is 530.929 cm².