A rock is thrown from the top of a tower that is 400 feet tall. If its height above the ground can be modeled by
h = -16t2 + 64t +400 where h is in feet, and t is in seconds.
To the nearest tenth of a second, how long does it take the rock to reach the ground?
To the nearest tenth of a second, how long will it take the rock to reach its maximum height?
To the nearest tenth of a foot, how high will the rock go before it starts falling down?
Hf=0=400+64t-16t^2
use the quadratic equation
a=-16, b=64, c=400
Now how high? Lets do some Gedunken.
if we find the time it takes for the rock to leave the tower, and return to the same altitute it started from
400=400+64t-16t^2 or
0=t(64-16t) the solutions are t=0, and t=4 seconds. Now the rock is at maximkum height in half that, or at t=2seconds
h=400+64t-16t^2 solve for t=2
I would like to point out a basic misunderstanding your teacher has with falling down. The rock is falling the instant is thrown upwards, that is why is slows down until the max height.
Satellites in earth orbit, albeit at constant altitude, are constantly falling, otherwise, they would go off in space.
To find the time it takes for the rock to reach the ground, we need to set h = 0 and solve for t.
Given that h = -16t^2 + 64t + 400, we have:
0 = -16t^2 + 64t + 400
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -16, b = 64, and c = 400. Substituting these values into the quadratic formula, we have:
t = (-64 ± √((64)^2 - 4(-16)(400))) / (2(-16))
Simplifying this expression, we get:
t = (-64 ± √(4096 + 25600)) / (-32)
t = (-64 ± √29696) / (-32)
t = (-64 ± 172.3) / (-32)
Using both the positive and negative square root results will give us two values. However, we only consider the positive square root value in this case because time cannot be negative. So, taking the positive square root:
t = (-64 + 172.3) / (-32)
t = 108.3 / -32
t ≈ -3.4 seconds (discarded)
Therefore, the time it takes for the rock to reach the ground is approximately 3.4 seconds.
To find the time it takes for the rock to reach its maximum height, we can determine the vertex of the parabolic equation h = -16t^2 + 64t + 400. The time at the vertex represents the time it takes for the rock to reach its maximum height.
The vertex of a quadratic equation in the form ax^2 + bx + c can be found using the formula:
t_vertex = -b / (2a)
In our equation, a = -16 and b = 64. Substituting these values, we have:
t_vertex = -64 / (2(-16))
t_vertex = -64 / (-32)
t_vertex = 2 seconds
Therefore, it will take the rock approximately 2 seconds to reach its maximum height.
To find the maximum height of the rock, we substitute the time (t_vertex) into the equation h = -16t^2 + 64t + 400:
h_max = -16(2)^2 + 64(2) + 400
h_max = -16(4) + 128 + 400
h_max = -64 + 128 + 400
h_max = 464 feet
Therefore, the rock will reach a maximum height of approximately 464 feet.
I hope this helps! Let me know if you have any other questions.