If a projectile is launched from a platform 30 feet high with an initial velocity of 128 feet per second, then the height of the projectile at t seconds is given by s(t) = –16t2 + 128t + 30 feet.
(a) At what time does the projectile attain its maximum height? Show some work.
(b) What is the maximum height attained by the projectile? Show some work.
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
(a) To find the time when the projectile attains its maximum height, we need to find the vertex of the quadratic equation. The vertex of a quadratic equation in the form of ax^2 + bx + c can be found using the formula -b/2a.
In this case, the equation is s(t) = -16t^2 + 128t + 30. Comparing it to the general form ax^2 + bx + c, we have a = -16, b = 128, and c = 30.
Using the formula, the time at which the projectile attains its maximum height can be found as:
t = -b/2a
= -128/(2*-16)
= -128/-32
= 4 seconds.
Therefore, the projectile attains its maximum height at t = 4 seconds.
(b) To find the maximum height attained by the projectile, we substitute the time (t = 4) into the equation s(t) = -16t^2 + 128t + 30.
s(4) = -16(4)^2 + 128(4) + 30
= -16(16) + 512 + 30
= -256 + 512 + 30
= 286 feet.
Therefore, the maximum height attained by the projectile is 286 feet.
To find the time at which the projectile attains its maximum height, we need to determine the vertex of the parabolic equation for the height of the projectile. The vertex form of a quadratic equation is given by:
s(t) = a(t - h)^2 + k
where (h, k) represents the coordinates of the vertex.
In the equation s(t) = –16t^2 + 128t + 30, we have a = -16, h = -b/2a, and k = s(h). Therefore:
h = -128 / (2 * -16) = 4
Now we substitute this value of h back into the equation to find the corresponding value of k:
k = s(4) = -16(4)^2 + 128(4) + 30 = 256 + 512 + 30 = 798
Therefore, the vertex of the equation is (4, 798).
(a) The time at which the projectile attains its maximum height is given by the x-coordinate of the vertex, which in this case is t = 4 seconds.
(b) The corresponding y-coordinate of the vertex represents the maximum height attained by the projectile. Therefore, the maximum height is 798 feet.