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

Question 1170787: 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 radius of the semicircle is r m, express the height of the rectangle in terms of r. Show that the area of the door has a maximum value when the radius
is 7/(4+π).
Found 2 solutions by math_helper, ikleyn:
Answer by math_helper(2461)   (Show Source): You can put this solution on YOUR website!

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.

Answer by ikleyn(52915)   (Show Source): You can put this solution on YOUR website!
.

For similar solved problem see the lesson
    - Finding the maximum area of the window of a special form,   Problem 2
in this site.

The similar/TWIN problem was solved there using Algebra only (without involving Calculus).

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this textbook under the topic "Finding minimum/maximum of quadratic functions".

You will find there many other classic problems on finding minimum/maximum of quadratic functions.

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.

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