Optimization problem with trig

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Homework Statement



An isosceles triangle has a rectangle inside of it with length 2 cm and width 6 cm. What angle ∅ will give the triangle the minimum area.

Homework Equations



A =1/2 (bh)

The Attempt at a Solution

 
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In your solution, what does x represent?
 
In my solution x represents the space which the rectangle does not cover.
Since the width of the rectangle is 6cm, and there are two spaces, The base of the triangle should be 6+2x.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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