Answer
To improve performance, the shot-putter should try to throw the object at an angle of $42^{\circ}$ and really focus on giving the object as much initial speed as possible.
Work Step by Step
As $\theta$ increases from $40^{\circ}$ to $42^{\circ}$, the distance increases slightly. As $\theta$ increases from $42^{\circ}$ to $45^{\circ}$, the distance decreases slightly. It seems that $42^{\circ}$ is close to the optimal angle. A change in the angle seems to have a small effect on the distance.
As $v$ increases from $43~ft/s$ to $45~ft/s$, the distance increases from $64.4 ~ft$ to $69.9 ~ft$. As $v$ increases, the distance increases significantly.
To improve performance, the shot-putter should try to throw the object at an angle of $42^{\circ}$ and really focus on giving the object as much initial speed as possible.
