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