Explanation:
a) To find the number of seconds that it takes for the rocket to reach the ground, we need to find the value of t when h = 0. We can set h = 0 and solve for t:
0 = 16t^2 + 96t + 10
To solve for t we need to use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/2a
Where:
a = 16, b = 96 and c = 10
t = (-96 ± √(96^2 - 41610))/2*16
t = (-96 ± √(9216 - 640))/32
t = (-96 ± √(8576))/32
t = (-96 ± 92.8)/32
So the rocket will take approximately 2.89 seconds to reach the ground.
b) To find the number of seconds that it will take the rocket to be at its highest maximum height, we need to find the value of t when the derivative of h is equal to 0.
h' = 32t + 96
h' = 0
32t + 96 = 0
t = -3
So the rocket will take approximately -3 seconds to be at its highest maximum height.
c) To find the maximum height of the rocket, we need to substitute the value of t from part b into the equation for h:
h = 16(-3)^2 + 96(-3) + 10
h = 169 + 96(-3) + 10
h = 144 - 288 + 10
h = -134
So the maximum height of the rocket is approximately -134 feet.