To find the time the pomegranate hits the ground, we need to find the value of t when the height is 0.
Given the function f(t) = 16t^2 + 128t, we can set it equal to 0 and solve for t:
16t^2 + 128t = 0
Factoring out a common factor of 16t:
16t(t + 8) = 0
Setting each factor equal to 0 and solving for t:
16t = 0, t + 8 = 0
From the first equation, we can see that t = 0.
From the second equation, we can see that t = -8.
Since time cannot be negative in this context, we can discard the solution t = -8.
Therefore, the pomegranate hits the ground at t = 0 seconds.
To find the time the pomegranate reaches its highest point, we need to find the vertex of the quadratic function f(t) = 16t^2 + 128t.
The vertex of a quadratic function in the form f(t) = at^2 + bt + c is given by t = -b / (2a).
In this case, a = 16 and b = 128.
t = -128 / (2 * 16) = -128 / 32 = -4
Therefore, the pomegranate reaches its highest point at t = -4 seconds.
Note that the time cannot be negative in this physical context, so we discard the solution t = -4.
Therefore, the pomegranate reaches its highest point at t = 4 seconds.