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?
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
1 Answer
"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,
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
(dA)/(d theta) = (pi-2theta)(cos theta) + (-2)(sin theta)
At a critical point (a minimum or a maximum) we require that the derivative,
(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
graph{(pi-2x)sin x [-2, 3, -2, 2.2]}
And we can verify that a maximum when