11. A rocket is fired upward from some initial height above the ground. Its height h in feet above the ground t seconds after it is fired is given by h = -16t2 + 48t +448.
(i) What is its initial height? __________ ft.
(ii) What is its maximum height? ___________ ft.
(iii) How long does it take to reach the ground? __________ sec.
-16t^2+48t+864
To find the initial height, we can simply substitute t = 0 into the given equation.
(i) Initial height: When t = 0, h = -16(0)^2 + 48(0) + 448 = 448 ft.
To find the maximum height, we need to determine the vertex of the parabolic equation. The maximum height occurs at the vertex.
(ii) Maximum height: The vertex of the parabolic equation h = -16t^2 + 48t + 448 can be found using the formula: t = -b/2a, where a = -16 and b = 48.
t = -(48) / (2*(-16)) = -48 / (-32) = 1.5
To find the maximum height, substitute t = 1.5 into the equation.
h = -16(1.5)^2 + 48(1.5) + 448
= -16(2.25) + 72 + 448
= -36 + 72 + 448
= 484 ft
(iii) To determine how long it takes for the rocket to reach the ground, we need to find the value of t when h = 0.
0 = -16t^2 + 48t + 448
16t^2 - 48t - 448 = 0
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, we'll use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
= (-(-48) ± √((-48)^2 - 4(16)(-448))) / (2(16))
= (48 ± √(2304 + 28672)) / 32
= (48 ± √(30976)) / 32
= (48 ± 176) / 32
Using the positive value since time cannot be negative:
t = (48 + 176) / 32 = 6 sec
Thus, it takes 6 seconds for the rocket to reach the ground.
To answer these questions, we need to understand the basic concepts of projectile motion and how to analyze equations representing the motion. Let's break it down step by step:
(i) What is its initial height?
The initial height of the rocket is the value of "h" when "t" is equal to 0. We can substitute the value of "t" in the equation and solve for "h."
Substituting t = 0 into the equation:
h = -16(0)^2 + 48(0) + 448
h = 448 ft
Therefore, the initial height of the rocket is 448 ft.
(ii) What is its maximum height?
The maximum height of the rocket can be found by determining the vertex of the quadratic equation. The vertex represents the highest point of the curve.
The equation h = -16t^2 + 48t + 448 is in the form of ax^2 + bx + c, where a = -16, b = 48, and c = 448.
The vertex of the quadratic equation is given by the formula:
t = -b/2a
Substituting the values of a and b into the formula:
t = -(48) / 2(-16)
t = -48 / -32
t = 1.5 sec
To find the maximum height (h), substitute the value of t into the equation:
h = -16(1.5)^2 + 48(1.5) + 448
h = -16(2.25) + 72 + 448
h = -36 + 72 + 448
h = 484 ft
Therefore, the maximum height of the rocket is 484 ft.
(iii) How long does it take to reach the ground?
To find the time it takes for the rocket to reach the ground, we need to solve the equation for h = 0. Set the height (h) equal to zero.
0 = -16t^2 + 48t + 448
This equation is a quadratic equation. We can solve for t by factoring, completing the square, or using the quadratic formula. In this case, we'll use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a),
where a = -16, b = 48, and c = 448.
Substituting the values into the quadratic formula:
t = (-48 ± √(48^2 - 4(-16)(448))) / (2(-16))
t = (-48 ± √(2304 + 28672)) / (-32)
t = (-48 ± √(30976)) / (-32)
t = (-48 ± 176) / (-32)
Simplifying further:
t = (48 - 176) / (-32) or t = (48 + 176) / (-32)
t = -128 / -32 or t = 224 / -32
t = 4 or t = -7
Since time cannot be negative in this context, we discard the negative value and consider t = 4 seconds.
Therefore, it takes 4 seconds for the rocket to reach the ground.
initial height is when t=0. 448
h = -16(t² - 3t - 28)
= -16(t-7)(t+4)
Note that h=0 when t=7. That's the answer to part iii
Since the two roots are -4 & 7, the apex is reached at t = (-4 + 7)/2 = 1.5
h(1.5) = -16(5.5)(5.5) = 484 ft