Question

a fireworks rocket is launched from a hill above a lake. the rocket will fall into the lake after exploding at its maximum height. the rockets height above the surface of the lake is given by $g(x) = -16x^2 + 64x + 80$. how long will it take the rocket to hit the lake?

Explanation:

Step1: Set height to 0 (lake level)

$g(x) = 0 \implies -16x^2 + 64x + 80 = 0$

Step2: Simplify the equation

Divide by $-16$: $x^2 - 4x - 5 = 0$

Step3: Factor the quadratic

$(x - 5)(x + 1) = 0$

Step4: Solve for x

$x - 5 = 0$ or $x + 1 = 0 \implies x = 5$ or $x = -1$

Step5: Discard negative time

Time cannot be negative, so $x = 5$

Answer:

5 units of time