Question

Challenge I came up with: what is the maximum area of the rectangle inscribed between the line `y=sintheta`, `x=theta` and `x=pi-theta`?

I conjured up this problem randomly, so I'm not sure if my wording is right. What is the maximum area for a rectangle inscribed in the curve `y=sinx`? What value of `theta` would there be for this?

Answer 1

`"Area" ~~ 1.12219267638209 ...`

Explanation:

Let us set up the following variables:

# { (x,"width of the rectangle"), (y, "height of the rectangle"), (A, "Area of the rectangle ") :} #

Our aim is to find an area function, `A(x,y,theta)` and eliminate the variables so that we can find the critical points wrt to the single variable `theta`.

The width of the rectangle is:

`x = (pi-theta)-(theta) = pi-2theta`

The Height of the rectangle is:

`y = sin theta`

The Area of the rectangle is given by:

`A = "width" xx "height"`
`\ \ \ = xy`
`\ \ \ = (pi-2theta)sin theta` ..... [*]

Differentiating wrt `theta`, using the product rule, we get:

`(dA)/(d theta) = (pi-2theta)(cos theta) + (-2)(sin theta)`

At a critical point (a minimum or a maximum) we require that the derivative, `(dA)/(d theta)` vanish, thus we require:

`(pi-2theta)cos theta -2sin theta = 0`
`:. 2sin theta = (pi-2theta)cos theta`

`:. 2sin theta/cos theta = pi-2theta`

`:. 2tan theta =pi-2theta`

We cannot solve this equation analytically and so we must use Numerical Techniques, the solution gainied is:

`theta ~~ 0.710462737775517...`

And so,

`A = (pi-2theta)sin theta`
`\ \ \ ~~ 1.12219267638209`

We need to establish that this value of `theta` corresponds to a maximum. This should be intuitive, but we can validate via a graph of the result [*] (or we could perform the second derivative test):
graph{(pi-2x)sin x [-2, 3, -2, 2.2]}

And we can verify that a maximum when `x~~0.7` of `~~1.1` is consistent with the graph.