SOLUTION: I need help: A door is in the shape of a rectangle surmounted by a semicircle whose diameter is equal to the width of the rectangle. If the perimeter of the door is 7 m, and the r

Let h be the height (length) of the door up to the bottom of the semicircle.

Let w be the width of the door



Notice right away, w = 2r



Perimeter = 7 =   =   



h as a function of r:



re-write the above:





divide both sides by 2:

   (*)





Now for maximum area: 





Recall w=2r and subs RHS of (*) found above.  We have the rectangle plus semicircle:

   



Simplify:

     (1)



Take derivative of A WRT r:





Set dA/dr to 0 to find critical point:





Solve for r:









One last thing: really should check to make sure we found a maximum (vs minimum).  One way to verify is to check values near  and make sure A is smaller on both sides. Another way is to graph A(r).  Another way, which I do here is to examine the 2nd derivative  at the critical point 

    



Since  the function (1) is concave down everywhere (no need to plug in for r) and the critical point is definitely a local MAXIMUM.