Answer:
Step-by-step explanation:
To find the sum of the time the projectile reaches its maximum height and the time it hits the ground, we need to determine the time values when the height function h(t) is equal to zero (ground level) and when the vertical velocity is zero (maximum height).
The height function of the projectile is given by:
h(t) = -16t^2 + 48t + 64
To find the time when the projectile hits the ground (height = 0), we set h(t) equal to zero:
0 = -16t^2 + 48t + 64
To simplify the equation, we can divide both sides by -16:
0 = t^2 - 3t - 4
Now, we can factor the quadratic equation:
0 = (t - 4)(t + 1)
Setting each factor equal to zero:
t - 4 = 0 ---> t = 4 seconds (time when the projectile hits the ground)
t + 1 = 0 ---> t = -1 seconds (discard this negative time in this context)
The time t = -1 seconds is not physically meaningful in this context, so we discard it.
Now, to find the time when the projectile reaches its maximum height, we know that the vertical velocity (v) is zero at the maximum height. The vertical velocity can be obtained by taking the derivative of the height function with respect to time (t):
v(t) = h'(t) = -32t + 48
To find the time when the vertical velocity is zero, we set v(t) equal to zero:
0 = -32t + 48
Solving for t:
32t = 48
t = 48 / 32
t = 1.5 seconds (time when the projectile reaches maximum height)
Now, the sum of the time the projectile reaches maximum height and the time it hits the ground is:
Sum of times = t_max_height + t_hit_ground
Sum of times = 1.5 seconds + 4 seconds
Sum of times = 5.5 seconds
Therefore, the sum of the time the projectile reaches maximum height and the time it hits the ground is 5.5 seconds