Question

Question #2041c

Answer 1

The maximum area is `128/(4 + pi) (~~ 17.92)`. This occurs when:

radius of the semicircle is `16/(4 + pi) (~~ 2.24)`
rectangular window is `16/(4+pi) xx 32/(4 + pi) (~~ 2.24 xx4.48)`

Explanation:

Let us set up the following variables:

`{(r, "Radius of the semicircle","(feet)"), (h, "Height of the rectangular window","(feet)"), (A, "Total area enclosed by the window", "(sq feet)") :}`

Our aim is to find `A(h,r)`, as a function of a single variable and to maximize the total area, `A`, wrt that variable (It won't matter which variable we do this with as we will get the same result). ie we want a critical point of `A` wrt the variable.

The total perimeter is that of `3` sides of the rectangle and the semicircle; we are told that this perimeter is `16` feet

`16 = (h + 2r + h) + (1/2)(2pir)`
`\ \ \ = 2r + 2h + pi r`

`:. 2h = 16 - 2r - pi r`
`:. \ \ h = 1/2(16 - 2r - pi r)`

And the total Area is that of a rectangle and a semicircle:

`A = (h)(2r) + (1/2)(pir^2)`
`\ \ \ = 2hr + 1/2 pi r^2`
`\ \ \ = 2(1/2(16 - 2r - pi r))r + 1/2 pi r^2`
`\ \ \ = 16r - 2r^2 - pi r^2 + 1/2 pi r^2`
`\ \ \ = 16r - 2r^2 - 1/2 pi r^2`

We now have the Area, `A`, as a function of a single variable `r`, so differentiating wrt `x` we get:

`(dA)/(dr) = 16 - 4r - pi r`

At a critical point we have `(dA)/(dr) =0 =>`

`16 - 4r - pi r = 0`
`:. \ \ \ \ 4r + pi r = 16`
`:. \ \ \ r(4 + pi) = 16`
`:. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \r = 16/(4 + pi) (~~ 2.24)`

With this value of `r` we have:

`A = 16(16/(4 + pi)) - 2(16/(4 + pi))^2 - 1/2 pi (16/(4 + pi))^2`
`\ \ \ = 128/(4 + pi) (~~ 17.92)`

And:

`h = 1/2(16 - 2(16/(4 + pi)) - pi (16/(4 + pi)))`
`\ \ = 16/(4+pi) (~~ 2.24)`

We can visually verify that this corresponds to a maximum by looking at the graph of `y=A(r)`:

graph{16x - 2x^2 - 1/2 pi x^2 [-5, 10, -5, 22]}

And also check that the perimeter is correct:

`P = 2r + 2h + pi r`
`\ \ = 2(16/(4 + pi)) + 2(16/(4 + pi)) + pi(16/(4 + pi))`
`\ \ = 16`