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?

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1 Answer
Feb 5, 2018

"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.