The height of a toy rocket in meters h(t) = -5t^2 + 4.5t + 8 where t is the time in secs. When is the height of the rocket more than 8ft above ground?

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mariloucortez | High School Teacher | (Level 3) Adjunct Educator

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The height of a toy rocket in meters h(t) = -5t^2 + 4.5t + 8 where t is the time in secs. When is the height of the rocket more than 8ft above ground?


First you need to convert 8ft to meters for unit consistency. Recall that 1m = 3.28ft.
So,

`8ft * (1m)/(3.28ft) = 2.43902439 m`

You are looking for the time when the rocket is above 8ft or 2.43902439m, so you can establish this equation.

`2.43902439 = -5t^2 + 4.5t + 8`

Combine similar terms. Move 8 on the right side. Note that the sign will change everytime you move a term from one side to the other.

`2.43902439 - 8 = -10t^2 + 4.5t`
`-5.56097561 = -5t^2 + 4.5t`

You can input `-5t^2 + 4.5t + 5.5609756 = 0` in your scientific calculator in the equation function to solve for the roots. You will have 2 values for t since it is a quadratic equation.

You'll get these values:
t = {1.596601553, -0.696601553}

Disregard the negative value since you can't have a negative value for time.

Therefore the time when the rocket is at 8 ft is 1.596601553s

When you try to get the maximum height by doing a derivative.

`h(t) = -5t^2 + 4.5t + 8`
`h'(t) = -10t + 4.5`

Equate it to 0.

`0 = -10t + 4.5`

move -10t to the other side to solve for t.

`10t = 4.5`

Divide both sides by 10.

`(10t)/10 = 4.5/10`

`t  = 0.45s`


At t = 0, you can solve that h(t) = 8m

When t = 0, it is already at 8 m. Notice that time at maximum is lesser than 1.596601553s. It means that the rocket is already going down.

So the time when the rocket is above 8ft is between t = 0 and t = 1.596601553s

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