Optimization Word Problem

Question

A building is to be built and will take the form of a semicircular shape and will have a rectangular doorway. The building will be 6 m long and the door must be at least 1 m vertically from the roof of the building.

A) What is the largest area the doorway can have and what are its dimensions(give the area to the nearest 0.01 m^2 and the dimensions to the nearest cm)

Answer 1

The problem isn't adequately worded. There isn't enough definition. Is the building a semi-circle shape looking from above or from an end view? How many degrees does the semi-circle cover? For example is it a half circle? 

 

Assuming a half circle (180 degrees) and that it is a semi circle from the end view, ie a long cylinder cut in half and laying down. Sort of shaped like a barn.

 

Let the diameter of the semi-circle be D. Let the door width be W and height H. 

 

Drawing a diagonal line from the bottom center of the door opening through the upper corner, the distance to the roof is 1 meter. The radius is D/2. So..

 

D/2 = 1 meter + sqrt [ H^2 + (W/2)^2 ]

solving for H:

  D/2 = 1 meter + sqrt [ H^2 + (W/2)^2 ]

  (D/2 - 1) = sqrt [ H^2 + (W/2)^2 ]

  (D/2 - 1)^2 = [ H^2 + (W/2)^2 ]

  H^2 = (D/2 - 1)^2 -  (W/2)^2

  H = sqrt [ (D/2 - 1)^2 - (W/2)^2 ]

 

Area = WxH.

       = W x sqrt [ (D/2 - 1)^2 - (W/2)^2 ]

 

To find the maximum, take the derivative and set to 0.

   d(Area)/dW = d{ W x sqrt [ (D/2 - 1)^2 - (W/2)^2 ] }/ dW = 0

   d{ W x sqrt [ (D/2 - 1)^2 - (W/2)^2 ] }/ dW = 0

   sqrt [ (D/2 - 1)^2 - (W/2)^2 ] + W(1/2) [(D/2 - 1)^2 - (W/2)^2]^-.5  {- 2 W/4 }= 0

 

notice the sqrt[] term appears in the denominator of the 2nd term.

Multiply the equation by the sqrt[] term:

   [ (D/2 - 1)^2 - (W/2)^2 ] + W(1/2) {- 2 W/4 }= 0

   [ (D/2 - 1)^2 - (W/2)^2 ] - W^2 /4= 0

 

   (D/2 - 1)^2 - (W/2)^2  - W^2 /4= 0

   (D/2 - 1)^2 - W^2/4 - W^2 /4= 0

   (D/2 - 1)^2 - W^2/2 = 0

 

    =>  W^2/2 = (D/2 - 1)^2

 

     W^2 = 2 (D/2 - 1)^2

     W = sqrt [2 (D/2 - 1)^2] for optimal door area.

      H = sqrt [ (D/2 - 1)^2 - (W/2)^2 ]