# 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#?

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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