Answer:
53.298°
Step-by-step explanation:
You want the maximum possible value of the angle θ in the given figure.
Angle
The angle θ is the sum of the angles at the top of the figure. For P = (x, 0), the angle at the top of the left shaded triangle is ...
arctan(x/7)
The angle at the top of the right shaded triangle is ...
arctan((5-x)/4)
The tangent of the sum of these two angles is found using the trig formula for the tangent of the sum of angles:
[tex]\tan(\theta_1+\theta_2)=\dfrac{\tan(\theta_1)+\tan(\theta_2)}{1-\tan(\theta_1)\tan(\theta_2)}\\\\\\\tan(\theta)=\dfrac{\dfrac{x}{7}+\dfrac{5-x}{4}}{1-\dfrac{x}{7}\cdot\dfrac{5-x}{4}}=\dfrac{35-3x}{x^2-5x+28}[/tex]
Maximum
The angle θ will be a maximum when the derivative of tan(θ) is zero.
[tex]\tan(\theta)'=0=\dfrac{(x^2-5x+28)(-3)-(35-3x)(2x-5)}{(x^2-5x+28)^2}\\\\\\\dfrac{3x^2-70x+91}{(x^2-5x+28)^2}=0\quad\Longrightarrow\quad x=\dfrac{35-2\sqrt{238}}{3}\approx 1.38183[/tex]
The value of θ for that value of x is found by evaluating the tan(θ) expression above:
[tex]\theta=\tan^{-1}\left(\dfrac{35-3(1.38183)}{(1.38183-5)1.38183+28}\right)=\tan^{-1}(1.34148)\\\\\\\theta\approx53.2975^\circ[/tex]
The maximum measure of θ is about 53.298°.
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Additional comment
We can't find a geometric construction that will locate point P. However, we observe that the location of P minimizes the radius of the circumcircle of the triangle containing θ.
