A rocket is shot into the air at a velocity of 48 ft/ sec. The equation is h(t) = -16t^2+48t+4

Question

1) How far off the ground did the rocket start from?
2) Using the quadratic function, how high the rocket will be at its maximum point?
3) How long will it take the rocket to reach that maximum point?
4) Rewrite the function into vertex form.
5) Predict when the rocket will reach the ground.
6) Sketch the function. ( Label at least 5 points.)

Answer 1

h(t) = a t² + (t) = -16 t² + 48 t + 4

The coefficients are:

a = - 16 , b = 48 , c = 4

1)

When rocket start t = 0

h(0) = -16 ∙ 0² + 48 ∙ 0 + 4 = 4 ft

2)

The vertex of a quadratic equation a t² + b t + c is the highest or lowest point of that equation.

If a < 0 the vertex of a quadratic equation is the highest point of that equation.

If a > 0 the vertex of a quadratic equation is the lowest point of that equation.

In this case a = - 16 < 0 so the vertex of a quadratic equation is the highest point of that equation.

The t-coordinate of the vertex is:

t = - b / 2 a

So t = - 48 / 2 ∙ ( - 16 ) = - 16 ∙ 3 / - 16 ∙ 2 = 3 / 2

h(max) = h ( 3 / 2 ) = -16 ∙ ( 3 / 2)² + 48 ∙ 3 / 2 + 4 =

-16 ∙ 9 / 4 + 144 / 2 + 4 = - 36 + 72 + 4 = 40 ft

3 )

You already calculated t coordinate of vertex.

It will take 3 / 2 = 1.5 sec

4)

The vertex form of a quadratic equation:

h = a ( t - h )² + k

where

h = t coordinate of vertex = 3 / 2

k = h(max) = 40

a = - 16

So the vertex form of this quadratic equation:

h = a ( t - h )² + k

h = - 16 ( t - 3 / 2 )² + 40

h = - 16 ( t - 1.5 )² + 40

5)

The rocket will reach the ground when h(t) = 0

For that you must solve:

-16 t² + 48 t + 4 = 0

Try to do it.

The solutions are:

3 / 2 - √5/2 = - 0.081139

and

3 / 2 + √5/2 = 3.081139

Time can not be negative so t = 3.081139 sec

6)

Go on wolframalpha. c o m

In rectangle paste:

plot -16 t² + 48 t + 4

then click option =

Answer 2

#1. plug in t=0
#2. as with all quadratics, the vertex is at t = -b/2a = 3/2
#3. See #2
#4. h = -16(t^2 - 3t + 9/4) + 4 + 36 = _____
#5. solve h=0