In the following exercise, consider a projectile launched at a height h feet abo

Question

In the following exercise, consider a projectile launched at a height h feet above the ground and at an angle ? with the horizontal. If the initial velocity is v0 feet per second, the path of the projectile is modeled by the parametric equations x = (v0 cos(?))t and y = h + (v0 sin(?))t ? 16t2. The center field fence in a ballpark is 10 feet high and 400 feet from home plate. The ball is hit h = 3 feet above the ground. It leaves the bat at an angle of ? degrees with the horizontal at a speed of 103 miles per hour. Find the minimum angle at which the ball must leave the bat in order for the hit to be a home run. (Round your answer to one decimal place.)

Explanation / Answer

s(t) = –16t^2 + 128t + 30 can do this with or without calculus using calculus ds/dt = -32 t + 128 = 0 at max height 32 t = 128 t = 4 seconds for max height s(4) = -16(16) +128(4) + 30 s(4) = 286 without calculus find vertex of parabola 16 t^2 -128 t = -s+30 complete square t^2 - 8t = -(1/16)s + (30/16) t^2 - 8 t + 16 = -(1/16)s + 15/8 + 16 (t-4)^2 = etc so vertex at t = 4

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