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Answer :
Final answer:
a. The rocket will take 8 seconds to return to the ground. b. The rocket will be 112 feet above the ground after 7 seconds. c. The rocket will hit its maximum height after 4 seconds. d. The maximum height of the rocket is 256 feet.
Explanation:
a. To find the time it takes for the rocket to return to the ground, we need to set the height to 0 and solve for t in the equation h(t) = -16t^2 + 128t. Setting h(t) to 0, we get 0 = -16t^2 + 128t. Factoring out a common factor of 16t, we have 0 = 16t(-t + 8). Setting each factor equal to 0, we find t = 0 and t = 8. The rocket will take 8 seconds to return to the ground.
b. To find the time it takes for the rocket to be 112 feet above the ground, we need to set h(t) equal to 112 and solve for t. The equation is 112 = -16t^2 + 128t. Rearranging the equation, we get 16t^2 - 128t + 112 = 0. Dividing by 16, we have t^2 - 8t + 7 = 0. Factoring the quadratic, we find (t - 7)(t - 1) = 0. This gives us two solutions, t = 7 and t = 1. However, t = 1 is not a valid solution as it is less than 0. Therefore, the rocket will be 112 feet above the ground after 7 seconds.
c. To find the time it takes for the rocket to hit its maximum height, we need to find the time at which the velocity is 0. The velocity of the rocket is given by v(t) = -32t + 128. Setting v(t) to 0, we have 0 = -32t + 128. Solving for t, we get t = 4. The rocket will hit its maximum height after 4 seconds.
d. To find the maximum height, we can substitute the time t = 4 into the equation h(t) = -16t^2 + 128t. h(4) = -16(4)^2 + 128(4) = -256 + 512 = 256. The maximum height of the rocket is 256 feet.
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Answer:
a. 8 seconds.
b. 1 second and 7 seconds.
c. 4 seconds.
d. 256 feet.
Step-by-step explanation:
The given function is
[tex]h(t) = -16t^{2}+128t[/tex]
The image attached shows the trajectory of the rocket in a height-time graph. Notice that the maxium point reached is (4, 256), which means after 4 seconds, the rocket has a maximum height of 256 feet. (c) and (d)
Additionally, a parabola is symmetrical, which means it takes the same time going up or down, therefore the rocket reaches the ground after 8 seconds. (a)
Now, when the rocket is 112 feet above the ground, we have
[tex]112 = -16t^{2}+128t[/tex]
Where we need to solve for [tex]t[/tex]
[tex]0 = -16t^{2}+128t-112[/tex]
Using a calculator, we have the solutions [tex]x_{1}= 1[/tex] and [tex]x_{2}=7[/tex], which means after 1 second and 7 second, the rocket is 112 feet above the ground. (b)
