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.
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?
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