A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 26 ft/s. At what rate is his distance from second base decreasing when he is halfway to first base?

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Luke Alvarez · Accepted Answer

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Samantha Adkison · Answer

undefined To find the rate at which the distance from the batter to second base is decreasing when he is halfway to first base, we need to use the chain rule from calculus. Let $ x $ be the distance between the batter and first base, and $ y $ be the distance between the batter and second base. The Pythagorean theorem gives us the relationship $ x^2 + y^2 = 90^2 $. Differentiating both sides with respect to time, we get $ 2x\frac{{dx}}{{dt}} + 2y\frac{{dy}}{{dt}} = 0 $. Given that $ \frac{{dx}}{{dt}} = 26 $ ft/s when $ x = 45 $ ft (halfway to first base), we can solve for $ \frac{{dy}}{{dt}} $. Plugging in the values, we find $ y = 45\sqrt{3} $ ft and $ \frac{{dy}}{{dt}} = -\frac{{13}}{{\sqrt{3}}} $ ft/s. Thus, the rate at which the distance from the batter to second base is decreasing when he is halfway to first base is approximately $ -7.5 $ ft/s.