PLEASE HELP ME!

Question

The function g(t) = -16t2 + vt + h represents the height of an object, g, in feet above the ground in relation to the time, t, in seconds, since the object was thrown into the air with an initial velocity of v feet per second at an initial height of h feet. A model rocket is launched off the ground at an initial velocity of 80 feet per second. What domain makes sense for the function in this context?

Answer 1

g(t) = height of the rocket at time t

domain should be from t=0 at liftoff until the time when it comes back and hits the ground

if it started at ground level, it hits the ground at twice the time it reaches max height

g'(t) = -32t + 80 = 0

t = 80/32 = 5/2 = 2 1/2 seconds

It hits the ground in 5 seconds. then the domain would be 0 to 5 or [0,5]

If it started at an unspecified height = h

g(t) = -16t^2 +80t + h = 0, solve for t, divide by -16t and complete the square

t^2 -5t = h/16

t^2 -5t + 25/4 = h/16 + 25/4 = (h+100)/16

t - 5/2 = + or - sqr(h+100)/16

t = 5/2 + or - (1/4)sqr(h+100)

t = 2 1/2 + or - (1/4)sqr(h+100)

domain then is [0, 2.5+(1/4)sqr(h+100]

that makes the domain interval larger than [0,5]