a model rocket is launched vertically into the air such that its height at any time, t, is given by the function h(t)=-16t^2+80t+10. What is the maximum height attained by the model rocket
Answers
Answer:
The first question asks how many seconds will go by before the rocket hits the ground. That means you're interested in when the rocket's height h(t) = 0. Factor the equation to find its zeroes:
h(t) = -16t2 + 128t
h(t) = t(-16t + 128)
0 = t(-16t + 128)
0 = t OR 0 = -16t + 128
0 = t OR -128 = -16t
0 = t OR -128/-16 = t
0 = t OR 8 = t
That means that at height = h(t) = 0, t = 0 or 8. At t = 0, zero seconds have gone by and the rocket hasn't yet left the ground, so t = 8 is when the rocket will return to the ground. That means the rocket will take 8 seconds to return to the ground.
To find out how many seconds it will take the rocket to reach 112 feet, plug in h(t) = 112, rearrange into quadratic form, and solve:
112 = -16t2 + 128t
0 = -16t2 + 128t - 112
0 = -16(t2 - 8t + 7)
0 = t2 - 8t + 7
0 = (t - 7)(t - 1)
0 = t - 7 OR 0 = t - 1
7 = t OR 1 = t
That means the rocket will be at height = h(t) = 112 at t = 7 and t = 1. That means the rocket will be 112 feet above the ground after 1 second and after 7 seconds.
The rocket will be at its max height halfway through the flight, since this is a parabola. We know the rocket lands at 8 seconds and launches at 0 seconds, so midway through the flight is (8+0)/2 = 8/2 = 4 seconds. That means the rocket will hit max height after 4 seconds.
To find the rocket's max height, plug in t = 4 and solve for h(4):
h(4) = -16(4)2 + 128(4)
h(4) = -16(16) + 512
h(4) = -256 + 512
h(4) = 256
The rocket will be at 256 at t = 4. That means that the rocket's max height is 256 feet.
Plz mark brainliest ❤️
Step-by-step explanation: