A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock’s height above ocean can be modeled by the equation H(t)=−16t2+96t+112.
a. When does the rock reach the maximum height?
b. What is the maximum height of the rock?
c. When does the rock hit the ocean?
a. To find the time when the rock reaches maximum height, we need to find the vertex of the parabolic function. The vertex of a parabolic function in the form y=ax^2+bx+c is given by the formula t=-b/(2a).
In this case, a=-16 and b=96.
t = -96/(2*-16) = -96/-32 = 3
Therefore, the rock reaches the maximum height after 3 seconds.
b. To find the maximum height of the rock, we can substitute t=3 into the equation H(t)=−16t2+96t+112.
H(3) = -16(3)^2 + 96(3) + 112 = -16(9) + 288 + 112 = 144
Therefore, the maximum height of the rock is 144 feet.
c. To find when the rock hits the ocean, we set H(t) = 0 and solve for t.
0 = -16t^2 + 96t + 112
0 = -t^2 + 6t + 7
0 = -(t-7)(t+1)
t=7 or t=-1
Since the time cannot be negative, the rock hits the ocean after 7 seconds.