A toy rocket is launched vertically from 5 feet above ground level with an initial velocity of 112 feet per second. The height h after t seconds is given by the equation: h(t) = -16t^2 + 112t + 5
How long will it take for the rocket to reach maximum height? What is the maximum height? and when will it return to the ground.
The vertex is 201.2 which would be the maximum height i think
You will have to find the vertex, there are several ways to do this.
1. Use Calculus: find the derivative , set it equal to zero.
that will give you the t of the vertex, sub it back in original to find the height h(t)
h' (t) = -32t + 112 = 0
t = 3.5
h(3.5) = -16(3.5)^2 + 112(3.5) + 5 = 201
so it will take 3.5 seconds to reach a max height of 201 ft
(the vertex is (3.5 , 201)
2. Complete the square. This method you should know, and you should get
h(t) = -16(t-3.5)^2 + 201
3. A good method to learn if you don't know Calculus is this:
for y = ax^2 + bx + c , the x of the vertex is -b/(2a)
so for yours, the t of the vertex = -112/(2(-16)) = 3.5
sub that back into the original and you got your vertex.
well, let me cheat to check
v = dh/dt = -32 t + 112 = 0 at top
so at top t = 112/32 = 3.5 seconds to top (vertex)
so h at top = -16t^2 + 112t + 5
= 5 + 112(3.5) - 16 (3.5)^2
= 201 at the top, ageed
well I told you 3.5 seconds to the top
now for the time when h = 0
0 = -16 t^2 + 112 t + 5
solve quadratic https://www.mathsisfun.com/quadratic-equation-solver.html
7.04 seconds (it only takes a fraction of a second to do that last 5 feet)
To find the time it takes for the rocket to reach its maximum height, we can use the equation h(t) = -16t^2 + 112t + 5. The maximum height corresponds to the vertex of the parabolic function.
The vertex of a parabola defined by the equation y = ax^2 + bx + c can be found using the formula x = -b / (2a). In this case, a = -16 and b = 112. Plugging these values into the formula, we have:
t = -112 / (2 * -16)
t = -112 / -32
t = 3.5 seconds
Therefore, it will take 3.5 seconds for the rocket to reach its maximum height.
To find the maximum height, we substitute the value of t back into the equation h(t):
h(3.5) = -16(3.5)^2 + 112(3.5) + 5
h(3.5) = -16(12.25) + 392 + 5
h(3.5) = -196 + 392 + 5
h(3.5) = 201 feet
So, the maximum height of the rocket is 201 feet.
To determine when the rocket will return to the ground, we need to find the time t when h(t) = 0. We can rearrange the equation to solve for t:
0 = -16t^2 + 112t + 5
16t^2 - 112t - 5 = 0
We can solve this quadratic equation using factoring, completing the square, or the quadratic formula. However, in this case, the quadratic equation has no real roots (no values of t that satisfy the equation). This means that the rocket will not return to the ground.
(Note: The reason for the rocket not returning to the ground in this equation is that the initial velocity is greater than the velocity required for the rocket to return to the ground. Therefore, the equation does not have any real roots.)