h=Vi*t - 16t^2=> 1st derivative of position(in this case h) with respect to time= velocity
dh/dt = Vi - 32t => max height is reached when velocity = 0
1500 - 32t = 0
t=1500/32 = 375/8 sec, time to reach max h. => sub. in the h equation to find max h
h = 1500*(375/8) - 16 (375/8)^2
h=35,156.25 ft => max h
The ascending and descending time are equal but let's prove it: set the h = 0
1500t - 16t^2 = 0 => solve the quadratic
t = 0 => reject
t = 375/4 seconds => time to hit the ground, notice that this is twice the time to reach max h
d = vt
t = d/v
= 50,000/880
≈ 56.8 sec to reach the flare but the flare reached max h at 375/8 ≈ 46.9 sec, so it won't hit the plane going up, but will it hit the plane on its way down from 35,156.25 ft which is 5,156.25 ft above the plane altitude? lol, I'll leave this one for you to think about.
part 2
Well the answer is a square. The side is 30 feet so the little goats will have 900 square feet in which to run. This is more than some people have to run in who are not very privileged.
To figure out it is a square, make the sides be H and L so you have 2H + 2L = 120 and A = H L
Then write the area in terms of a single variable, A = H (120 - 2H ) / 2 = 60 H - H^2 since L = (120 - 2H ) / 2
Then take the derivative, dA/dH = 60 - 2H
Set it to zero to find the critical points
60 - 2H = 0
Solve for H
H=30
