Part 1:
A missile is shot upward from a submarine 1440 feet below sea level at an initial velocity of 528 ft/sec. A function that approximates the height of the missile (relative to sea level) is given by:
h(t) = -16t^3 + 528t - 1440
where h(t) is the height of the missile in feet and t is the time in seconds.
Complete the table below to determine the height of the missile for the given values of time, t.
time (t)
0=
5=
15=
25=
35=
Part 2:
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.
Part 1: To determine the height of the missile at different time intervals, we can substitute the given values of time (t) into the function h(t) = -16t^3 + 528t - 1440.
First, let's substitute the value of 0 for t into the function:
h(0) = -16(0)^3 + 528(0) - 1440
= 0 + 0 - 1440
= -1440
So, when t = 0, the height of the missile is -1440 feet.
Now, let's substitute the value of 5 for t into the function:
h(5) = -16(5)^3 + 528(5) - 1440
= -16(125) + 2640 - 1440
= -2000 + 2640 - 1440
= 200
So, when t = 5, the height of the missile is 200 feet.
Similarly, we can calculate the heights at t = 15, 25, and 35 by substituting these values into the function:
h(15) = -16(15)^3 + 528(15) - 1440
= -16(3375) + 7920 - 1440
= -54000 + 7920 - 1440
= -46520
h(25) = -16(25)^3 + 528(25) - 1440
= -16(15625) + 13200 - 1440
= -250000 + 13200 - 1440
= -238240
h(35) = -16(35)^3 + 528(35) - 1440
= -16(42875) + 18480 - 1440
= -686000 + 18480 - 1440
= -668960
So, the height of the missile at t = 15 is -46520 feet, at t = 25 is -238240 feet, and at t = 35 is -668960 feet.
Part 2: To find the time required for the missile to emerge from the water and reenter the water, we need to find the values of t when the height of the missile is equal to 0.
The function is h(t) = -16t^3 + 528t - 1440. To find the time when the missile emerges from the water, we set h(t) equal to 0:
0 = -16t^3 + 528t - 1440
We can start by factoring out the GCF (Greatest Common Factor) of the terms, which is 16:
0 = 16(-t^3 + 33t - 90)
Now, we can solve the remaining quadratic equation (-t^3 + 33t - 90) by factoring or using the quadratic formula.
Unfortunately, the cubic equation doesn't factor neatly, so we need to use the cubic formula or approximate the zeros using numerical methods.
Once we find the values of t when h(t) = 0, we can determine the time required for the missile to emerge from the water and reenter the water.
Please note that further calculations are required to determine the exact values of t when the missile emerges from and reenters the water.
Your equation should say
h(t) = -16t^2 + 528t - 1440 , not cubed
Besides that, it would have that function only after it leaves the water.
It would be different in the water.
So I will let you fix that