Question

model rocket is launched from a raised platform at a speed of 76 feet per second. Its height in feet is given by h(t)=-16t^2+76t+24 (t=seconds after launch) What is the maximum height reached by the rocket? and what time does it meet maximum height? And how long is the rocket in the air?​

Answer 1

Hello!

This is a question relating a quadratic equation to the vertex and roots.

Since this parabola has a negative a value, the vertex will be the maximum height that this rocket reaches.

We can find the vertex (h,k) with the following equations.

`h=(-b)/(2a)`

`k=h(h)`

`h=(-76)/(2(-16))`

`h=(76)/(32)`

`h=2.375`

`k=-16(2.375)^2+76(2.375)+24`

`k=114.25`

We can interpret the values like this:

At
`t=2.375` seconds after the rocket was launched, the rocket reached its maximum height of
`114.25` feet.

Since the y-intercept is at
`(0,24)` and this is a negative parabola, there will only be one positive root, which will be how long our rocket is in the air.

Use the quadratic formula.

`x=(-b+-√(b^2-4ac))/(2a)`

`x=(-(76)+-√((76)^2-4(-16)(24)))/(2(-16))`

`x=5.407, -0.297`

Since we are searching for our positive root, this rocket is in the air for 5.407 seconds.

Hope this helps!