5 Steps to a 5: AP Calculus BC 2021 by William Ma

Author:William Ma
Language: eng
Format: epub
Publisher: McGraw-Hill Education
Published: 2021-02-15T00:00:00+00:00

The left field fence in Boston’s Fenway Park, nicknamed the Green Monster, is 37 feet high and 310 feet from home plate. If a ball is hit 3 feet above the ground and leaves the bat at an angle of , write a vector-valued function for the path of the ball and use the function to determine the minimum speed at which the ball must leave the bat to be a home run. At that speed, what is the maximum height the ball attains?

Step 1: The horizontal component of the ball’s motion, the motion in the “x” direction, is . The vertical component follows the parabolic motion model , where g is the acceleration due to gravity. The path of the ball can be represented by the vector-valued function .

Step 2: In order for the ball to clear the fence, its height must be greater than 37 feet when its distance from the plate is 310 feet. , solved for t, gives seconds. At this time, , and this value must exceed 37 feet. Setting and solving gives s ≈ 105.556. The ball must leave the bat at 105.556 feet per second in order to clear the wall.

Step 3: Since , the derivative is , and the ball will attain its maximum height when the vertical component is equal to zero. Since s ≈ 105.556, produces t ≈ 2.462 seconds. For that value of t, ≈ 183.762, 89.779. The ball will reach a maximum height of 89.779 feet, when it is 183.762 feet from home plate.

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