How to Find the Largest Area and Dimensions of a Rectangle in a Semi Circle

Question

A rectangle is to be inscribed in a semicircle of a radius of 7 cm as shown in the following figure.

Note: The angle shown in the image is θ.

(a) Find the function that models the area of the rectangle.

(b) Find the largest possible area for such an inscribed rectangle. [Hint: Use the fact that sin(u)

 achieves its maximum value at u = 𝜋/2.]

(c) Find the dimensions of the inscribed rectangle with the largest possible area. (Round your answers to two decimal places.)

Answer 1

Let x = half the width and y = the height of the rectangle

Area, A = 2xy

x = √(49-y²)

sin(θ) = y/7 --> y = 7sin(θ)

Substituting for x and y

A = 2√(49-49sin²(θ))(7sin(θ))

= 2√(49(1-sin²(θ)) (7sin(θ))

= 2(7)(7)sin(θ)√cos²(θ)

= 49 (2sin(θ)cos(θ))

= 49sin(2θ)

A will be its max when 2θ = π/2 or θ = π/4

y = 7 sin(π/4) = 7/√2

tan(π/4) = 1 = y/x --> x = y = 7/√2

The dimensions of the rectangle are x = 7/√2 cm and y= 7/√2 cm (This is a square)

The maximum area is, A = 2(7/√2)(7/√2) = 49 cm²