Area of multi-shape window

Question

A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal to the width of the rectangle. If the perimeter of the window is 742 cm, find maximum area of the window in order to admit the most light and the dimensions of the window.

 

A) max area?

B) Length of the base of the window

C) total height of window including rectangle and the semicircle 

Answer 1

For a rectangle the perimeter is 2H + 2W (H and W being the height and width).

For a circle the Perimeter (Circumference) is Pi•D.

 

For this window the D is equal to the W of the rectangle

and the rectangular part has only 3 sides

and the circular part is 1/2 the circumference of the circle.

 

So the total perimeter of the window is

2H + W + (Pi•W)/2 = 742 cm

This is one equation.

 

The area of the rectangle part is = H•W

The area of the semi-circle is = 1/2 [Pi r^2] = 1/2 [Pi•(W/2)^2]

The total area is then:

H•W + 1/2 [Pi•(W/2)^2] = Amax

This is the other equation.

 

Two equations, two unknowns, H & W

Solve the first for H in terms of W

Plug that H into the second equation

(as if you were solving by substitution)

Then solve the second for the value that gives the largest area.