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Start by letting L = 4r. You then have a function in two unknowns. The maximum will be at points where the partial derivatives are 0 and the Hessian is negative definite, or if you can show the function is concave at that point.

Originally Posted by charlottewill
Please could somebody help me with this tutorial equation

The velocity (v) of a piston is related to the angular velocity (ω) of the crank by the relationship v = ωr{sinθ+(r/(2L))(sin2θ)} where r is length of crank and L is length of connecting rod. Find the first positive value of θ for which v is a maximum, for the case when L = 4r.

Regards,

Charlotte
$\displaystyle \frac{dv}{d\theta}=\omega r \cdot \left(\cos \theta \sin\2 \theta +2 \cos 2\theta \left(\sin \theta +\frac{r}{2L}\right)\right)$

$\displaystyle \frac{dv}{d\theta}=0 \Rightarrow \cos \theta \sin\2 \theta +2 \cos 2\theta \left(\sin \theta +\frac{r}{2L}\right)=0$

for $\displaystyle L=4r$ you have :

$\displaystyle \cos \theta \sin\2 \theta +2 \cos 2\theta \left(\sin \theta +\frac{1}{8}\right)=0$