Mock Exams AP Calculus BC – FRQ Set 3

Question

The Boston Red Sox play in Fenway Park, notorious for its Green Monster, a wall 37 feet tall and 315 feet from home plate at the left-field foul line. Suppose a batter hits a ball 2 feet above home plate, driving the ball down the left-field line at an initial angle of $30^{\circ}$ above the horizontal, with initial velocity of 120 feet per second. (Since Fenway is near sea level, assume that the acceleration due to gravity is $-32.172 \mathrm{ft} / \mathrm{sec}^2$.)

(a) Write the parametric equations for the location of the ball $t$ seconds after it has been hit.

(b) At what elevation does the ball hit the wall?

(c) How fast is the ball traveling when it hits the wall?

▶️Answer/Explanation

(a) The following table shows x- and y-components of acceleration, velocity, and position:

The last line in the table is the answer to part (a).
(b) To determine how far above the ground the ball is when it hits the wall, find out when $x=315$, and evaluate $y$ at that time.
$
\begin{aligned}
& 60 \sqrt{3 t}=315 \text { yields } t=\frac{315}{60 \sqrt{3}} \approx 3.03109 \text { seconds } \\
& y\left(\frac{315}{60 \sqrt{3}}\right) \approx 36.075 \text { feet }
\end{aligned}
$
(c) The ball’s speed at the moment of impact in part (b) is $|v(t)|$ evaluated at $t=\frac{315}{60 \sqrt{3}}$.
$
\begin{aligned}
& |v(t)|=\sqrt{\left(v_x(t)\right)^2+\left(v_y(t)\right)^2} \\
& =\sqrt{(60 \sqrt{3})^2+(-32.172 t+60)^2} \text { when } x=315 \\
& \left|v\left(\frac{315}{60 \sqrt{3}}\right)\right| \approx 110.487 \mathrm{ft} / \mathrm{sec}
\end{aligned}
$

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