What does the 24t mean in the context of the situation?

Question

The equation f(t) = - 16t ^ 2 + 24t + 8 represents the height of a bottle rocket launched into the air based on t, the number of seconds the rocket is in the air . The graph of the flight is shown below.

Graph Link: https://drive.google.com/file/d/1YJ2p-xOD5QFXwbh6L5vUCWEi8KIhsoQG/view?usp=drivesdk

A. The initial velocity of the bottle rocket

B. The initial height of the bottle rocket

C. The amount of time that the bottle rocket is in the Air

D. The highest height the bottle rocket reached on its flight path

Answer 1

24 is the initial velocity. 24t is the distance traveled at time t. at t=0 it didn't go anywhere. time t = 1 it traveled 24 feet but minus the effect of gravity, the t^2 term.

A) 24 feet per second is the initial velocity

B) 8 feet is the initial height

C) time in the air is found by setting h=0 and solving for t

-16t^2 +24t + 8 = 0

2t^2 -3t -1 = 0

try to factor, it doesn't factor, but almost which suggests a possible sign typo in the problem, but accepting it as given

t= 3/4 + sqr(9+8) = 3/4+sqr17 = 4.873 seconds in the air

D) max height is found by setting the derivative = 0, which leads to t=3/4

plug 3/4 into the original equation to get h=-16(3/4)^2 + 24(3/4) + 8= 17 feet for maximum height. It reached max height of 17 feet after 3/4 of a second