There's a point $A(4t\sin\theta,4t\cos\theta)$ (where $\theta$ is a constant),at moment of $T=t$,constantly moving.
Also Let there be points $B_1,B_2,B_3$whose coordinates are$B_1(10,0),B_2(-5,5\sqrt3),B_3(-5,-5\sqrt3)$
when $T=0$.
These points start to chase moving point $A$ since $t=0$.
point $B_1,B_2,B_3$ will satisfy these conditions;
A) Point $B_2,B_3$ will move at speed of $v$,while point $B_1$ will move at speed of $\frac{v}{2}$.
B) At any given moment $T=t$ for $t \geq 0$, point $B_{1,2,3}$ will be heading to $A$.(so point $B_{1,2,3}$'s direction changing will be occuring continuously.)
Let $t_c (t_c \geq 0)$ be the minimum value of $T$ that satisfies
$A=B_1$ or $A=B_2$ or $A=B_3$(In other words, point $B_1$ or $B_2$ or $B_3$ 'caught' $A$ when $T=t_c$.)
(Note:$T$ represents time, while $t$ is variable.)
(1)Find the value of $\theta$ which maximizes $t_c$ when $v=10$.
(2)Find the range of $v$ if the value of $t_c$ doesn't exist for some $\theta.$(doesn't exist means $t_c$ being negative.)
I've tried to solve with implicit differentiation, but i don't know how to solve this; Can you help it? I'll be appreciating your help.