Parametric Equations: Maximum

Question

Find the time it takes for a ball following the path of the parametric equations, x = 4t - 8 and y = -t^2 + 18t, to reach a maximum

Answer 1

y = 18t - t^2

y' = 18 -2t = 0

t = 18/2 = 9 seconds to reach max height

Answer 2

We're reaching a maximum in terms of y, so we just need to find the vertex of the y function and plug that back into the x function to see where the point is. The vertex (h, k) can be found easily with h = ^-b/_2a = ^-18/_2(-1) = 9. This makes the maximum at -9² + 18*9 = 81 for t=9 and thus y=81. Then plugging t=9 into the x function, we get x = 4*9 - 8 = 28.

Therefore, we have a maximum when t=9 and it occurs at the point (28, 81).