Question

Amelia launched her science fair rocket, and then backed away. The path of the rocket is given by the equation below, where y is the the height in feet amd x is the time in seconds the rocket is in the air y=-2×^2+5x+7

1)when does the rocket hit the ground?
2)what is the maximum height of the rocket?

Answer 1

1)The rocket hit the ground at
`x = (5)/(4)`

2)The maximum height of the rocket = 12.468 feet

Explanation:

Step(i):-

Given equation

y = -2 x² + 5 x + 7 ...(i)

Differentiating equation (i) with respective to 'x' , we get

`(dy)/(dx) = -2( 2x) +5(1) = -4x +5`

Equating zero

`(dy)/(dx) = 0`

⇒ -4 x +5 =0

⇒ -4 x = -5

⇒
`x = (5)/(4)`

The rocket hit the ground at
`x = (5)/(4)`

Step(ii):-

`(dy)/(dx) = -4x +5` ...(ii)

Again differentiating equation (ii) with respective to 'x' , we get

`(d^(2) y)/(dx^(2) ) = - 4(1) <0`

The maximum height at x =
`(-5)/(4)`

y = -2 x² + 5 x + 7

`y = -2 ((5)/(4))^(2) + 5 ((5)/(4) ) + 7`

`y = (-2(25)+ 25 (16)+7(64))/(64)`

`y = (798)/(64) = 12.468 feet`

The maximum height of the rocket = 12.468 feet