A model rocket is launched straight upward with initial velocity of 22 m/s. The height of the rocket, h, in metres, can be modelled by h(t) = -5t^2 + 22t, where t is elapsed time in seconds. What is the maximum height the rocket reaches?
To find the maximum height the rocket reaches, we need to determine the vertex of the quadratic function representing the height of the rocket. The vertex of a quadratic function in the form h(t) = a*t^2 + b*t + c is given by the formula:
t = -b / (2*a)
In this case, a = -5 and b = 22. Let's calculate the time at which the rocket reaches its maximum height:
t = -22 / (2*(-5))
t = -22 / (-10)
t = 2.2
Therefore, the rocket reaches its maximum height after 2.2 seconds.
To find the height at this time, substitute t = 2.2 into the height equation:
h(2.2) = -5*(2.2)^2 + 22*(2.2)
h(2.2) = -5*4.84 + 22*2.2
h(2.2) = -24.2 + 48.4
h(2.2) = 24.2
Therefore, the maximum height the rocket reaches is 24.2 meters.
To find the maximum height the rocket reaches, we need to determine the vertex of the quadratic equation h(t) = -5t^2 + 22t.
The vertex of a quadratic equation in the form h(t) = at^2 + bt + c is given by the coordinates (t, h) where t = -b/2a and h = h(t).
In this equation, a = -5 and b = 22. Let's substitute these values into the formula for t:
t = -b/2a
t = -22/(2*(-5))
t = -22/(-10)
t = 11/5
t = 2.2
Now we can find the maximum height by substituting t = 2.2 into the equation for h:
h(2.2) = -5(2.2)^2 + 22(2.2)
h(2.2) = -5(4.84) + 48.4
h(2.2) = -24.2 + 48.4
h(2.2) = 24.2
Therefore, the maximum height the rocket reaches is 24.2 meters.
Easy if you know calculus.
h ' (t) = -10t + 22 = 0 for a max/min of h(t)
10t = 22
t = 2.2
h(2.2) = -5(2.2)^2 + 22(2.2) = appr 24.2 m
If you don't know Calculus, the x of the vertex of your parabola is -b/(2a)
= -22/-10 = 2.2
sub in 2.2 into h(t) , same as above
If you don't know about the -b/(2a) stuff, you will have to complete the square to find the vertex.