Question

A basketball player is shooting a basketball toward the net. The height, in feet, of the ball t seconds after the shot is modeled by the equation h = 6 + 30t – 16t2. Two-tenths of a second after the shot is launched, an opposing player leaps up to block the shot. The height of the shot blocker’s outstretched hands t seconds after he leaps is modeled by the equation h = 9 + 25t – 16t2. If the ball was on a path to reach the net 1.7 seconds after the shooter launches it, does the leaping player block the shot?
yes, exactly 0.6 seconds after the shot is launched
yes, between 0.64 seconds and 0.65 seconds after the shot is launched
yes, between 0.84 seconds and 0.85 seconds after the shot is launched
no, shot not blocked

Answer 1

Let T be seconds after the shooter launches. The ball’s height is
h_ball = 6 + 30T − 16T^2.

The blocker jumps 0.2 s later, so his time variable is t = T − 0.2. His hand height as a function of T is
h_block = 9 + 25(T − 0.2) − 16(T − 0.2)^2
= 3.36 + 31.4T − 16T^2.

Set h_ball = h_block:
6 + 30T − 16T^2 = 3.36 + 31.4T − 16T^2.
The −16T^2 terms cancel, leaving 6 + 30T = 3.36 + 31.4T,
so 2.64 = 1.4T and T = 2.64/1.4 ≈ 1.8857 s.

That intersection occurs at about 1.886 s, which is later than 1.7 s (when the ball reaches the net). Therefore the blocker does not block the shot.

Answer: no, shot not blocked.