A rocket follows a parabolic trajectory. After t seconds, the vertical height of the rocket above the ground, in metres, is given by H(t) = 37t – t . Find the length of time the rocket is in the air.
Could you please explain me, why answer is 37s?
Thank you so, so much:))
I think you mean
H(t) = 37t-t^2
H(t) = 0 at t=0 and t=37
So, at t=0 the rocket took off (height=0) and at t=37 it came back down (H=0 again). Elapsed time = 37.
Well, first of all, I have to apologize because there seems to be a mistake in the given equation. It should be H(t) = 37t - t^2, not H(t) = 37t - t.
Now, let's find the length of time the rocket is in the air. We know that the rocket will be on the ground when its height, H(t), is equal to zero. So, we'll set the equation equal to zero and solve for t:
0 = 37t - t^2
To solve this quadratic equation, we can use factoring or the quadratic formula. But since it's a bit more fun, let's try factoring:
0 = t(37 - t)
To satisfy this equation, either t = 0 or (37 - t) = 0.
If t = 0, it means the rocket was on the ground at the start. But since we want to find the time the rocket is in the air, we can ignore this solution.
If (37 - t) = 0, then t = 37.
So, the length of time the rocket is in the air is 37 seconds.
To find the length of time the rocket is in the air, we need to determine when the vertical height of the rocket above the ground is equal to zero. This signifies the moment when the rocket hits the ground.
Given the equation for the height of the rocket above the ground as H(t) = 37t - t^2, we can set this equation equal to zero to find the time when the rocket reaches the ground:
37t - t^2 = 0
To solve this quadratic equation, we can factor it as:
t(37 - t) = 0
From this equation, we can see that the solutions for t are when t = 0 or when 37 - t = 0.
When t = 0, the rocket is just starting its trajectory, so it hasn't reached the ground yet. Therefore, we can ignore this solution.
When 37 - t = 0, solving for t gives us t = 37. This value represents the time it takes for the rocket to hit the ground.
Thus, the length of time the rocket is in the air is 37 seconds.
To find the length of time the rocket is in the air, we need to determine when the vertical height, H(t), becomes zero. Setting H(t) = 0 and solving for t will give us the answer.
Given: H(t) = 37t - t
Setting H(t) = 0:
37t - t = 0
Combining like terms:
36t = 0
Dividing both sides by 36:
t = 0
So, the rocket is in the air for t seconds, where t = 0.
However, this does not make sense as the rocket needs to be in the air for some time. Therefore, there seems to be a mistake in the given equation for H(t).
If we assume that the correct equation for H(t) is H(t) = -37t + t^2, we can find the length of time the rocket is in the air:
Setting H(t) = 0:
-37t + t^2 = 0
This is a quadratic equation, so we can solve it by factoring:
t(-37 + t) = 0
From this equation, we can see that the rocket will be in the air for two different times: t = 0 and t = 37.
However, since time cannot be negative, the rocket is only in the air for t = 37 seconds.
Therefore, the correct answer is 37 seconds, not 0 seconds.
Note: It is important to double-check equations and their given information to ensure accurate solutions.