Find the length of the base of a Norman window

Question

A Norman window has the shape of a rectangle capped by a semicircle. What is the length of the base of a Norman window of maximum area if the perimeter of the window equals 60?

Answer 1

We're going to want to find an expression for the area in terms of the length of the base and maximize it through differentiation.

 

Start by defining the variables.  Let l = the length of the base and let h be the height.  From the description, that means that l/2 is the radius of the semicircle.

 

The area of the window will be the sum of the areas of the rectangle and the semicircle.  That's lh + π(l/2)² / 2 = lh + πl² / 8

 

That's a function of the area in terms of length and height, so we want to get rid of the height somehow.  We can do that by using the fact that the perimeter is 60.

 

The perimeter will include the bottom base, the two sides and the arc of the semicircle.  That's 2h + l + 2π (l/2) / 2 = 2h + l + πl/2.

 

Now let's use that to find h in terms of l:

 

60 = 2h + l + πl/2

60 - l - πl/2 = 2h

60 - l(1 + π/2) = 2h

[60 - l(1 + π/2)] / 2 = h

30 - l (1/2 + π/4) = h

 

Now go back to the expression for the area and substitute this in for h:

 

A = lh + πl² / 8

A = l [30 - l (1/2 + π/4) ] + πl² / 8

A = 30l -1/2 l² - π/4 l² + πl² / 8

A = 30l + l² (π/8 - 1/2 - π/4)

A = 30l + l² (-π/8 - 1/2)

A = 30l - l² (π/8 + 1/2)

 

Now differentiate A wrt l:

dA/dl = 30 - 2l (π/8 + 1/2)

 

Set the derivative to 0 and solve for l:

30 - 2l (π/8 + 1/2) = 0

30 = 2l (π/8 + 1/2)

15 = l (π/8 + 1/2)

15 / (π/8 + 1/2) = l

 

l is approximately equal to 16.8

 

To approximate h (in case we're interested):

60 = 2h + l + πl/2

60 = 2h + l (1 + π/2)

60 = 2h + 16.8 * 2.57

60 = 2h + 43.18

16.82 = 2h

8.41 = h

 

Please review my calculations to make sure you agree, or find a correction.