A model rocket is shot straight into the air. Its height in meters a "t" seconds is given by h=-4.9^2+29.4t.

a) What is the rocket's initial height?
c)When does the rocket reach the ground?

h = -4.9t^2 + 29.4t

for the initial height, t = 0 , so h = 0

"reach the ground" implies h = 0
-4.9t^2 + 29.4t = 0
t(-4.9t + 29.4) = 0
so t = 0 ----> the initial time and height
or
t = 29.4/4.9 = 6

so it will return to the ground after 6 seconds

To find the initial height of the rocket (h₀), we have to plug in t = 0 into the equation h = -4.9t^2 + 29.4t.

a) Initial height (h₀) when t = 0:
h₀ = -4.9(0)^2 + 29.4(0)
h₀ = 0 + 0
h₀ = 0

Therefore, the rocket's initial height is 0 meters.

To find when the rocket reaches the ground, we have to find the value of t when h = 0.

c) When the rocket reaches the ground (h = 0):
0 = -4.9t^2 + 29.4t

Now we can solve this quadratic equation. Firstly, let's set it equal to zero:
-4.9t^2 + 29.4t = 0

Next, we can factor out a common factor of t:
t(-4.9t + 29.4) = 0

Now, we can set each factor equal to zero and solve for t:

t = 0 (one possibility)

-4.9t + 29.4 = 0
-4.9t = -29.4
t = -29.4 / -4.9
t = 6

Therefore, the rocket reaches the ground after 6 seconds.

To find the rocket's initial height, we can look at the equation h = -4.9t^2 + 29.4t. The initial height corresponds to the value of h when t = 0.

a) To find the initial height, we substitute t = 0 into the equation:
h = -4.9(0)^2 + 29.4(0)
h = 0

Therefore, the rocket's initial height is 0 meters.

To find when the rocket reaches the ground, we need to determine the value of t when the height is 0. In other words, we need to solve the equation -4.9t^2 + 29.4t = 0.

c) To find when the rocket reaches the ground, we can factor out a common factor of t from the equation:
t(-4.9t + 29.4) = 0

Now we have two possibilities for t:
1) t = 0, which means the rocket is already on the ground (the initial time)
2) -4.9t + 29.4 = 0 (which gives us the time when the rocket will reach the ground)

To find the value of t in the second case, we solve the equation -4.9t + 29.4 = 0 for t:
-4.9t = -29.4
t = -29.4 / -4.9
t = 6

Therefore, the rocket reaches the ground after 6 seconds.