the path of a rocket fired during a fireworks display is given by the equation s(t)=64t-16t^2 (t to the second power) where t is the time, in seconds, and s is the height, in feet. what is the maximum height, in feet, the rocket will reach? In how many seconds will the rocket hit the ground?

Question

the path of a rocket fired during a fireworks display is given by the equation s(t)=64t-16t^2 (t to the second power) where t is the time, in seconds, and s is the height, in feet. what is the maximum height, in feet, the rocket will reach? In how many seconds will the rocket hit the ground?

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3 answers
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Answer 1

bobpursley answered
14 years ago

This is a parabola. Zeroes (that is s(t) is zero ) when t=0, and t=4.

So the max height due to symettry of the parabola occurs at t=2.

Answer 2

Explain Bot answered
1 year ago

To find the maximum height the rocket will reach, we need to find the vertex of the parabolic equation s(t) = 64t - 16t^2. The vertex of a parabola in the form of y = ax^2 + bx + c is given by the formula:

t = -b / (2a)

In this case, a = -16 and b = 64. Let's substitute these values into the formula:

t = -(64) / (2(-16))
t = -64 / (-32)
t = 2

So, the rocket will reach its maximum height at t = 2 seconds.

To find the maximum height, we substitute the value of t back into the equation s(t):

s(t) = 64t - 16t^2
s(2) = 64(2) - 16(2^2)
s(2) = 128 - 16(4)
s(2) = 128 - 64
s(2) = 64

Therefore, the rocket will reach a maximum height of 64 feet.

To find the time it takes for the rocket to hit the ground, we need to find the value of t when s(t) = 0. The equation for s(t) is already given as s(t) = 64t - 16t^2. Let's set it equal to zero and solve for t:

64t - 16t^2 = 0
t(64 - 16t) = 0

From this equation, we can determine that either t = 0 or 64 - 16t = 0.

For t = 0, the rocket hasn't been fired yet, so it is not the solution we are looking for.

To find the value of t when 64 - 16t = 0, we can solve for t:

64 - 16t = 0
16t = 64
t = 64 / 16
t = 4

Therefore, the rocket will hit the ground after 4 seconds.

In summary:
- The maximum height the rocket will reach is 64 feet.
- The rocket will hit the ground after 4 seconds.

Answer 3

Step-by-Step Bot answered
1 year ago

To find the maximum height the rocket will reach, we need to determine the vertex of the parabolic equation. The vertex form of a parabola equation is given by y = a(x - h)^2 + k, where (h, k) represents the vertex.

In this case, the equation s(t) = 64t - 16t^2 represents the height of the rocket. Therefore, we can rewrite it as s(t) = -16t^2 + 64t.

Comparing this with the vertex form, we can see that a = -16, which means the parabola opens downwards. The vertex is given by the formula:
h = -b/2a
where b is the coefficient of the linear term (-16t).

So, substituting the values, we have:
h = -64 / 2*(-16)
h = -64 / -32
h = 2

Therefore, the rocket will reach its maximum height of 2 feet.

To find the time it takes for the rocket to hit the ground, we need to determine when the height reaches zero. We can set s(t) = 0 and solve for t.

-16t^2 + 64t = 0

Factoring out common terms, we get:
-16t(t - 4) = 0

Using the zero product property, we can set each factor equal to zero:

-16t = 0 or t - 4 = 0

Solving for t, we have:
t = 0 or t = 4

Therefore, the rocket will hit the ground in 4 seconds.