Question

How do you maximize a window that consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches?

Answer 1

found two values, for the width and height (of the rectangular part) of your window:

80.6

in and

40.4

in

Explanation:

Considering your window as:

enter image source here

Perimeter is:

P

=

2

a

+

b

+

π

(

b

2

)

=

288

so:

a

=

1

2

[

288

−

b

−

π

2

b

]

(1)

Area is:

A

=

b

⋅

a

+

π

2

(

b

2

)

2

using the value of

a

from (1):

A

=

b

1

2

[

288

−

b

−

π

2

b

]

+

π

2

(

b

2

)

2

=

=

144

b

−

b

2

2

−

π

4

b

2

+

π

8

b

2

=

144

b

−

b

2

(

1

2

+

π

4

−

π

8

)

Maximize the area deriving it and setting it equal to zero:

A

'

=

144

−

2

b

(

1

2

+

π

4

−

π

8

)

=

0

so that

b

=

80.6

in

So from (1):

a

=

40.4

in

Where I used for the semicircle:

Perimeter

=

1

2

⋅

2

π

r

Area

=

1

2

⋅

π

r

2