A model rocket is launched from the roof of a building?

Method 1 to find the maximum height : Completing square

h = -5t² + 30t + 10
h = -(5t² - 30t) + 10
h = -5(t² - 6t + 9) + 10 + 5*9
h = 55 - 5(t - 3)²

For all real values of t, -5(t - 3)² ≤ 0
Hence, h = 55 - 5(t - 3)² ≤ 55
Maximum h = 55 when t = 3

The maximum height = 55 m


Method 2 to find the maximum height : Differentiation

h = -5t² + 30t + 10
h' = -10t - 30 = -10(t - 3)
h" = -10

When t = 3 :
h = -5(3)² + 30(3) + 10 = 55
h' = 0
h" = -10 < 0

Hence, the maximum height = 55 m


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To find the time taken for the rocket to land.

When the rocket lands, h = 0
-5t² + 30t + 10 = 0
(-5t² + 30t + 10) / (-5) = 0
t² - 6t - 2 = 0
t = {-(-6) ± √[(-6)² - 4(1)(-5)]| / 2
t = 6.3 or t = -0.32 (rejected)

Time taken for the rocket to land = 6.3 s

i) h'(t) = -10t+30 = 0 for maximum
t = 3 seconds
h(3) = 55 m
ii) 0 = -5t^2+30t+10
t^2-6t-2 = 0
t = [6±√44]/2
t = 3+√11 seconds
≈ 6.317 seconds
See Graph
https://www.desmos.com/calculator/l4jnmiipko

dh/dt = -10t + 30
-10t + 30 = 0
10t = 30
t = 30/10
t = 3
h = -5*3^2 + 30*3 + 10
h = -5*9 + 90 + 10
h = -45 + 90 + 10
h = 55

h=-5t²+30t+10
At max height change in height is 0. This means dh/dt=0
dh/dt=-10t+30
0 = -10t+30
10t=30
t=3
h(3) = -5(3)²+30(3)+10 = -45+90+10 = 55 units
The max height is 55 length units

On landing h=0
0=-5t²+30t+10
t² - 6t - 2 = 0
Using quadratic solution
t = [-(-6)±√((-6)²-4(1)(-2))]/2(1)
t = (6±√44)/2
t = 6.3 s
the rocket takes 6.3s to land

h = -5t² + 30t + 10

The height function you give is consistent with a free-falling projectile. It is not at all consistent with a rocket. That is the condition though, so go with the quadratic function.

The time of the extreme value can be had from the coefficients.

t = -30 / [2(-5)] = 3

Substitute t = 3 and solve for h, the maximum height.

The "rocket" was launched from a height of 10 meters. If it lands on the same rooftop, the height will again be 10. In that case, solve this quadratic equation, and reject the zero solution:

h = 10
-5t² + 30t + 10 = 10

If it lands on the ground, its height will then be zero. If that is the case, then solve this quadratic equation, and reject the negative solution:

h = 0
-5t² + 30t + 10 = 0