# Long distance revolution

Calculus Level 4

For a long time, NBA basketball was defined by aggressive rim to rim play, a volley of short shots, interrupted by the occasional long distance heave. In the past few years, the Golden State Warriors discovered something — the three pointer is worth fifty percent more than the two point shot. Today, the Warriors have transformed the game with their three point shooting, and have such dominance over other teams that some analysts have suggested that the three point line be moved back.

For the average league shooter, success decreases with distance to the rim. However, the Warrior's shooters (led by Steph Curry) actually improve with distance. Suppose that the field goal percentage of the average shooter as a function of feet from the rim, $$x$$, is given by the fit $$l(x)$$ and that the Warriors' shooters are described by $$w(x)$$. If the NBA wants to minimize the relative advantage of the Warriors, where should they place the line?

Assumptions and Details

• $$l(x) = 37 - \dfrac15 (x-23)^2$$
• $$w(x) = 43 + \dfrac37 (x-26)^2$$
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