A rocket follows a parabolic trajectory. After t seconds, the vertical height above the ground, in metres, is given by H=37t - t (squared).
What is the maximum height of the rocket after 10 seconds?
What is the max height of the rocket?
What time does it reach the maximum height?
How long is the rocket in the air for?
this doesnt make any sense wtd is dh
Ah, a rocket on a parabolic trajectory, how exciting! Let's answer your questions one by one, shall we?
1. To find the maximum height of the rocket after 10 seconds, we need to substitute t with 10 in the equation H = 37t - t^2. That gives us H = 37(10) - (10)^2. Solving that, we get H = 370 - 100 = 270 meters.
2. The maximum height of the rocket is 270 meters. Quite an impressive height, I must say!
3. To determine the time at which the rocket reaches its maximum height, we need to look at the equation H = 37t - t^2. Since this is a parabolic trajectory, the maximum height is reached when the coefficient of the t^2 term changes from positive to negative. In this case, it happens at t = 18.5 seconds. So, the rocket reaches its maximum height at 18.5 seconds.
4. Ah, the duration the rocket stays airborne, a crucial question! Since we know the rocket reaches its maximum height at 18.5 seconds, we can deduce that it spends an equal amount of time going up and coming back down. So, multiplying the time it takes to reach maximum height (18.5 seconds) by two gives us the total time the rocket is in the air: 18.5 x 2 = 37 seconds.
Hope that helps, and happy rocket watching! Let me know if you have any more questions to launch at me!
To find the maximum height of the rocket after 10 seconds, we can substitute t = 10 into the equation H = 37t - t^2:
H = 37(10) - (10^2)
H = 370 - 100
H = 270 meters
Therefore, the maximum height of the rocket after 10 seconds is 270 meters.
To find the maximum height of the rocket, we need to find the vertex of the parabola. The equation H = 37t - t^2 can be written in the form H = -t^2 + 37t.
The vertex of the parabola is given by the formula t = -b / (2a), where a is the coefficient of t^2 and b is the coefficient of t.
So, in this case, a = -1 and b = 37.
t = -37 / (2 * -1)
t = -37 / -2
t = 18.5
So, the rocket reaches its maximum height at t = 18.5 seconds.
To find how long the rocket is in the air for, we need to determine the time it takes for the object to hit the ground. Since the rocket's height above the ground is given by a quadratic equation, we can set it equal to zero and solve for t:
0 = 37t - t^2
t^2 - 37t = 0
t(t - 37) = 0
So, the rocket spends either 0 seconds (at the beginning) or 37 seconds (when it hits the ground) in the air.
Therefore, the rocket is in the air for 37 seconds.
To find the maximum height of the rocket, we need to determine the vertex of the parabola. The equation for the vertical height, H, of the rocket is given by H = 37t - t^2.
First, let's find the maximum height after 10 seconds.
To do that, plug in t = 10 into the equation for H:
H = 37(10) - (10)^2
H = 370 - 100
H = 270 meters
So, the maximum height of the rocket after 10 seconds is 270 meters.
Now, let's find the time at which the rocket reaches its maximum height.
The vertex of a parabola in the form y = ax^2 + bx + c is given by the x-coordinate x = -b/2a.
In our equation for H, H = 37t - t^2, the coefficient of t^2 is -1, and the coefficient of t is 37. Therefore, a = -1 and b = 37.
To find the time at the vertex, we substitute these values into the equation x = -b/2a:
t = -37 / 2(-1)
t = -37 / -2
t = 18.5 seconds
So, the time at which the rocket reaches its maximum height is 18.5 seconds.
Finally, let's determine the duration the rocket stays in the air.
Since the rocket reaches its highest point at 18.5 seconds and gravity pulls it back down, the rocket's total flight time is twice the time it takes to reach the maximum height.
Therefore, the rocket is in the air for 2 * 18.5 = 37 seconds.
h = 37 t - t^2
at t = 10
h = 370 - 100 = 270
dh/dt = 0 at top = 37 - 2t
t = 37/2 = 18.5 seconds to top
h = 37(18.5) -18.5^2 = 342
when is h = 0 again?
t(t-37) = 0
t = 37 seconds
(which of course is 18.5 times two)