Question

I throw a stone off a cliff 32 feet above the water.  The height of the stone in terms of time (in seconds) is given by h(t) = -16t2 + 128t + 32.  At what time will the stone splash into the water? Round your answer to nearest tenth of a second

Answer 1

Your equation for the height of the stone at any time is   h(t) = -16t2 + 128t + 32 .

From your equation, we can tell that you're defining the upward direction as
positive.  We can also tell that you threw the stone upward, with an initial speed
as it left your hand of 128 feet per second, about 87 miles per hour ... a mighty toss indeed, and I think there's a man from the Chicago Cubs waiting outside
who'd like to talk to you.

Anyway, When the stone splashes into the water,  h(t) = 0 .

-16t² + 128t + 32 = 0

Divide each side by -16 :

t² - 8t - 2 = 0

I don't see any easy way to factor the expression on the left,
so I have to use the quadratic formula to solve this equation.

t = 4 plus and minus √18 .

t = +8.24 seconds
t = -0.24 second

Mathematically, both numbers are valid solutions.But when you apply
the equation to a real world situation, only the positive 't' makes sense.
So   t = 8.24 seconds.