Question

Please help me with this math question!!

Answer 1

The rocket splashes down after
`\boxed{24}` seconds

The rocket peaks at
`\boxed{769}` meters above sea-leval

Explanation:

The relationship between height and time is given by the equation

`h(t) = -4.9t^2+112t+129`

The splashdown will occur when
`h(t) = 0`

Therefore solve t for

`-4.9t^2+112t+129 = 0`

This is a quadratic equation which can be solved with the aid of a quadratic formula calculator

The solution set for this equation is

`t=-1.09894 \;and\;\:t=23.95609\right)`

Omitting the negative value we get

`\text{t=23.95609 \;seconds}\\\\= 2\text{4 \;seconds \;rounded}}`

For the second part, the maximum value of h(t) can be found by finding the first derivative h'(t) and setting it equal to 0 and solving for t; then plug this value of t into h(t) to find the max height

`h'(t) = (d)/(dt)\left(-4.9t^2+112t+129\right)\\\\= (d)/(dt)\left(4.9t^2\right)+(d)/(dt)\left(112t\right)+(d)/(dt)\left(129\right)\\\\= - 9.8t + 112 + 0\\\\= -9.8t + 112\\\\`

Setting the above expression to 0 gives

`-9.8t+112=0\\\\-9.8t = -112\\\\t = (-112)/(-9.8)\\\\t = 11.4286 \;seconds`

Plugging this value back into h(t) gives

`h(11.4286) = -4.9\cdot \:11.4286^2+112\cdot \:11.4286+129\\\\= 769 \;meters \;(rounded)`