Substituting and solving multiple steps

Question

Part 1) Please solve each equation in order to get the height for each one. h(0)=-16t2+528t-1440 h(5)=-16t2+528t-1440 h(15)=-16t2+528t-1440 h(25)=-16t2+528t-1440 h(35)=-16t2+528t-1440 Part 2) Interpret the meaning of EACH negative value you came up with in the previous question. Describe the behavior of the missile at each of the times. Part 3) Using the same steps as you followed in the first question of this case study, find the time required for the missile to emerge from the water and the time required for the missile to reenter the water. Start your factorization by taking out the GCF. (Hint – The height of the missile, h(t)=0 at sea level). This is a 10-point question so be sure you are showing ALL work. As always, write a sentence that gives your answer including the behavior of the missile at those times.

Answer 1

h(t) = -16t^2 +528t- 1440

h(0) = -1440 feet= 1,440 below sea level, = initial height at time t=0

h(5)= -16(5^2) + 528(5) -1440 = -400 + 2640 -1440 = -400 + 1200 = 800 feet high after 5 seconds

h(15) = -16(225) + 528(15) -1440 = 3600 + 7920 -1440 = 3600 + 6480 = 10,080 feet high in 15 seconds

h(25) = -16(25)^2 + 528(25) -1440 = -16(625) + 13,200 -1440 = 23,200 -1440 = 21,760 ft high in 25 seconds

h(35) = -16(35)^2 + 528(35) -1440 = -16(1225) + 18,480 - 1,440 = -19,600 + 17,040 = - 2,560 ft in 35 seconds. It's falling back down into the water

h(t) = -16t^2 +528t -1440 = -16(t^2 - 33t + (33/2)^2) -1440 + (33/2)^2

= -16(t- 33/2)^2 - 1440 + 1089/4

= -16(t - 16.5)^2 - 4671/4

= -16(t- 16.5)^2 - 1167.75 is vertex form with vertex = maximum point (h,k) = (16.5 sec, 1167.75 ft)

maximum height = 1,167 3/4 ft after 16.5 seconds

after 16 1/2 seconds, it begins to fall back to the water

from t= 0 to 16 1/2 it's rising higher

h(t) =0 = -16(t - 16.5)^2 =- 1167.75

-16(t-16.5)^2 =

(t-16.5)^2 = 1167.75/64

t-16.5 = +/-(sqr1167.75)/2

t = 16.5 +/- (sqr1167.75)/2 = when the missile leaves the water and re-enters the water

= about 16.5 +/-17.086

= - 0.586 and 33.586 seconds