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Optimization Word Problem

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)

I spend almost 4 hours trying to do this question, but I don't understand how to use the 1 m in my working out!

1 Answer by Expert Tutors

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Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...
4.9 4.9 (639 lesson ratings) (639)
This question is a bit loosely worded, so some aspects are not completely clear.  What follows is my best guess as to what is envisioned by this problem.
It looks like we have a building with one  semicircular  outer wall into which a door is cut.  (Architects would call this the elevation view).
The radius of the semicircle is  6/2 =  3m.
Each point on the edge of semicircle can be described by the coordinate pair (x, sqrt(9 - x2) ) .
The 9 is the square of 3 which is the radius.  x is the horizontal distance from the origin.  The origin is located in the center at ground level.  The y coordinate is the height above ground.
The straight forward approach is to maximize the area =  2x sqrt(9 - x2) with respect to x.   Taking the derivative and setting to zero results in  x = sqrt(3)  and y =  sqrt(6).   However, sqrt(6) = 2.45.  This would lead to 3 - 2.45 vertical clearance from the max height (3) of the wall.  This is less than the 1 m required.  The conclusion from this is that the constraint ( 1m clearance) is in effect a critical point for the problem.
At this new critical point  y =2    (i.e. 3 - the 1 clearance).  
So  2 =  sqrt(9 -x2)   and x =  sqrt(5)
The area is  2 x sqrt(9 -x2) =  4 sqrt(5)    and the dimensions are
2sqrt(5)  by 2.